ring extensions, injective covers and envelopes

Ring extensions, injective covers and envelopes

By

DAVID DEMPSEY, LUIS OYONARTE 1) and YEONG-MOO SONG 2)

Abstract. The present paper is devoted to the study of those rings R such that for anyring homomorphism R! S the functor HomR�S;ÿ� : R-Mod! S-Mod preservesinjective envelopes or injective covers.

The case of injective envelopes has been studied by T. Würfel ([9]), who gave acharacterization of such rings (Theorem 10). In this paper we give anothercharacterization of those rings in Section 2. One of the tools we use is a generalizationof a certain type of module initially studied by Northcott ([4]), McKerrow ([3]) and Park([5] and [6]).

The case of injective covers is treated in Section 3, where we give a completecharacterization of commutative noetherian rings satisfying the property mentioned above.

1. Preliminaries. Throughout this paper every ring will be associative with identity. For anyring R, not necessarily commutative, the symbol Z�R� will denote the center of R. If N; Mare two modules, we will write N % M to say that N is a submodule of M, and N % 0M to saythat N is an essential submodule of M.

Recall that an injective envelope of a module M is an essential monomorphism ofR-modules M ,! E with E an injective R-module, and that injective envelopes are unique upto isomorphisms. Therefore, the injective envelope of a module M will be denoted as E�M�.

Given any R-module M and any ideal I of R, we recall that depthIM is defined asdepthIM � minfi; Exti�R=I;M� �j 0g; and Ass�M� is the set of all prime ideals P of R suchthat P � Ann�x� for some element x of M.

The next definition can be found, for example, in [7, Chapter VIII, §2].

De f i n i t i on 1 . 1 . For any module M and any ordinal number a, soca�M� is defined asfollows: soc0�M� � 0, soca�M�=socaÿ1�M� � soc M=socaÿ1�M�ÿ �

whenever a is a successorordinal and soca�M� � [

b<asocb�M� if a is a limit ordinal.

Now we introduce the concept of semi-artinian modules and rings, which have beenwidely studied by different authors.

De f i n i t i on 1 . 2 . A nonzero module M is said to be semi-artinian if there exists an ordinalnumber a for which soca�M� �M. A ring R is right semi-artinian if it is as a right R-module.

Arch. Math. 76 (2001) 250±2580003-889X/01/040250-09 $ 3.30/0� Birkhäuser Verlag, Basel, 2001 Archiv der Mathematik

Mathematics Subject Classification (1991): 16D10, 16E30, 13E05.1) The author was supported by a grant from F.P.U., Subprograma General de Perfeccionamiento de

Doctores en el Extranjero, and partially supported by the grant CRG 971543 from NATO.2) The author was partially supported by Sunchon National University, 1998.

The next proposition characterizes the semi-artinian rings. For a proof see for example[7, Proposition 2.5, Chapter VIII].

Proposition 1.3. The following properties of a ring R are equivalent:

i) R is right semi-artinian.ii) Every right R-module is semi-artinian.

iii) Every right R-module is an essential extension of its socle.

2. Ring extensions and injective envelopes. A first example of a ring R such that thefunctor HomR�S;ÿ� does not always preserve injective envelopes is any integral domainwhich is not a field, as we can deduce from the following argument.

If R is any ring and R ,! S is any ring extension with HomR�S;ÿ� preserving injectiveenvelopes, then for any R-module M �j 0 we have HomR�S;M� �j 0 sinceHomR�S;E�M�� j 0 (note that HomR�R;E�M�� E�M� �j 0 and so HomR�S;E�M�� j 0)and HomR�S;M� is an essential S-submodule of HomR�S;E�M��.

Therefore if I is any proper left ideal of R, then IS �j S since IS � S impliesHomR�S;R=I� � 0, which is absurd from the above (R=I �j 0).

Now if R is any integral domain which is not a field and K is its field of fractions, for anyproper ideal I of R we have IK � K, so for the extension R ,! K the functor HomR�K;ÿ�cannot preserve injective envelopes.

The proof of the following lemma is straightforward so we will omit it.

Lemma 2.1. Let R be any ring (commutative or not). If for any left R-module N, everyessential submodule M % 0N and any index set I, MI is an essential submodule of NI, thenQi2I

Mi % 0Qi2I

Ni for any index set I, whenever Mi % 0Ni for any i 2 I.

Let fxi; i 2 Ig be an infinite set of elements and let R Xh i denote the semigroupalgebra with X the free semigroup on the elements xi; i 2 I. Given any left (right)R-module M, define M Xÿ1

� �to be the module of formal power series on the

noncommuting indeterminates xÿ1i , i 2 I, with coefficients in M, i.e., the module consisting

of those m� Pn ^ 1

P�i1 ;...;in�2In

mi1;...;in xÿ1i1 � � � xÿ1

in �m� Pn ^ 1

P�i1;...;in�2In

xÿ1i1� � � xÿ1

in mi1;...;in� with

m;mi1 ;...;in 2M. It is immediate to verify that this left (right) R-module can be seen as aleft (right) R Xh i-module with the following structure; to describe such an R Xh i-modulestructure it suffices to define scalar multiplication for ªmonomialsº of the formrxj1 � � � xjn 2 R Xh i and mxÿ1

i1 � � � xÿ1im 2M Xÿ1

� �. So the product of the last two monomials

is defined as rxj1 � � � xjn �mxÿ1i1 � � � xÿ1

im � rmxÿ1in�1� � � xÿ1

im whenever jn � i1; jnÿ1 � i2; . . . ; j1 � inand 0 otherwise (and the right R Xh i-module structure is thus defined in the obvious way).

A straightforward computation shows that if M is a left R-module, then the mapf : HomR R Xh i;M� � !M Xÿ1

� �given by

f�f � � f �1� � Pn ^ 1

P�i1;...;in�2In

f �xi1 � � � xin� xÿ1in � � � xÿ1

i1

!is an isomorphism of left R Xh i-modules. Analogously, if M is a right R-module and we

251Vol. 76, 2001 Ring extensions, injective covers and envelopes

define f by

f�f � � f �1� � Pn ^ 1

P�i1;...;in�2In

xÿ1in� � � xÿ1

i1 f �xi1 � � � xin � !

;

then f is an isomorphism of right R Xh i-modules.Constructions of this type have been studied, for example, by D. G. Northcott in [4], for

the case of commuting indeterminates and R �M � K a field, and more generally byA. S. McKerrow in [3] and S. Park in [5] and [6].

A similar construction for the case when the elements of R do not commute with theindeterminates xi is due to Roger D. Warren in [8], where he proved the following: If G isany �R;R�-bimodule and T � R�G��GR G��GR GR G�� . . . is the tensoralgebra, then given any ring homomorphism f : R! S and an �R;R�-linear maph : G! S, there exists a unique ring homomorphism g : T ! S such that gjR � f andgjG � h ([8, Theorem 1]).

We then see that if G � RZ R, the tensor algebra on G can be considered, in a certainway, as a ring of polynomials in one indeterminate x � 1 1, where we do not assumexr � rx for r 2 R. To make the same construction for more indeterminates we just have toreplace RZ R by a direct sum of copies of RZ R.

Therefore, if we have any ring homomorphism f : R! S, we see that if wechoose G � RZ R� �� RZ R� �� . . . large enough, f can be factored throughR ,! R�G��GR G�� . . . ÿ!g S where g is surjective. For more information about thistopic see [8].

Now, it is well known that any ring homomorphism having the form R! R=I preservesinjective envelopes, and any surjective R! S is essentially R! R=I. This implies thesurjective g mentioned above preserves injective envelopes.

On the other hand, given R!g S!h A, where HomR�S;ÿ� and HomS�A;ÿ� both preserveinjective envelopes, we have that the morphism f � h � g is such that HomR�A;ÿ� preservesinjective envelopes, since HomS�A;HomR�S;ÿ�� HomR�AS S;ÿ� � HomR�A;ÿ�:

Therefore to prove that HomR�S;ÿ� preserves injective envelopes for any ringhomomorphism R! S, we only have to prove that HomR�T;ÿ� preserves injectiveenvelopes when T � R�G��GR G��GR GR G�� . . ., where G � �I�R�Z R� forany index set I.

Theorem 2.2. Let R be a ring (not necessarily commutative). For any ring homomorphismR! S the functor HomR�S;ÿ� preserves injective envelopes of left (right) R-modules if andonly if i) and ii) below hold.

i) If I is any index set and fMi; i 2 Ig, fNi; i 2 Ig are two families of left (right)R-modules with Mi % 0Ni for all i 2 I, then

Qi2I

Mi % 0Qi2I

Ni.

ii) HomR�RZ R;M� % 0HomR�RZ R;N� as left (right) R-modules whenever M is anessential R-submodule of N.

Note that condition ii) is equivalent to:

iii) For any essential left (right) R-submodule M of N, the left (right) R-moduleHomZ�R;M� is essential in HomZ�R;N�.

252 D. DEMPSEY, L. OYONARTE and Y.-M. SONG ARCH. MATH.

P roof. Let us assume that HomR�S;ÿ� preserves injective envelopes of left (right) R-modules for any ring homomorphism R! S.

i) By hypothesis, if we let X be the free semigroup on the elements xi; i 2 I for anarbitrary index set I, the map HomR�R Xh i;M� ! HomR�R Xh i;N� is an essentialmonomorphism of left R Xh i-modules for any essential left R-submodule M of N. ButM Xÿ1 � �

is isomorphic to HomR�R Xh i;M� as left R Xh i-modules, thus M Xÿ1 � �

is anessential left R Xh i-submodule of N Xÿ1

� �. If we prove that M Xÿ1

� �% 0N Xÿ1

� �also

as R-modules, we will have thatQ

n2N

�QIn

M�

is an essential R-submodule ofQ

n2N

�QIn

N�

, so

MI % 0NI (they are direct summands ofQ

n2N

�QIn

M�

andQ

n2N

�QIn

N�

respectively). Therefore,

by Lemma 2.1Qi2I

Mi % 0Qi2I

Ni for any index set I, whenever Mi % 0Ni for any i 2 I.

Let a � a0 �P

n ^ 1

PIn

ai1;...;in xÿ1i1 � � � xÿ1

in be an element of N Xÿ1 � �

. Let us prove that there

exists r 2 R such that r � a �j 0, r � a 2M Xÿ1 � �

, so then M Xÿ1 � �

% 0N Xÿ1 � �

asR-modules. With this purpose we will construct an element a � a0 �

Pn ^ 1

PIn

ai1;...;in xÿ1i1 � � � xÿ1

in

of N Xÿ1 � �

with the following properties: 1) every coefficient of a is either zero or acoefficient of a (i.e., it is equal to a coefficient of a), and 2) for any coefficient ai1;...;it of athere exists an infinite number of n 2 N such that ai1;...;it � aj1 ;...;jnÿt ;i1;...;it for all�j1; . . . ; jnÿt; i1; . . . ; it� 2 In.

In order to define a, we will define each one of its coefficients, proceeding as follows:We let a0 be a coefficient of a (specified below), let ai1 � 0 8i1 2 I, and let ai1 ;i2 be a

coefficient of a (specified below) for all �i1; i2� 2 I2. For any n 2 N, n ^ 2, if ai1 ;...;in is acoefficient of a, then ai1;...;ij � 0 for all �i1; . . . ij� 2 Ij and all j such that n < j < 2n, and ai1 ;...;i2n

is again a coefficient of a (specified below) for all �i1; . . . ; i2n� 2 I2n.We now specify the coefficients of a which are coefficients of a: a0 � a0, ai1;i2 � a0

8�i1; i2� 2 I2 and ai1 ;...;i4 � ai4 8�i1; . . . ; i4� 2 I4 (thus, we see that every coefficient of a of theform ai is also a coefficient of a). We then start over and define ai1;...;i8 � a0 8�i1; . . . ; i8� 2 I8,ai1;...;i16 � ai16 8�i1; . . . ; i16� 2 I16 and ai1 ;...;i32 � ai31;i32 8�i1; . . . ; i32� 2 I32. We proceed byinduction on n getting that if ai1;...;in � a0 for all �i1; . . . ; in� 2 In and some n 2 N, and ifai1;...;in=2 � ail ;...;in=2 (note that n=2 is a natural number since n must be even by construction)for all �i1; . . . ; in=2� 2 In=2, where l � n=2ÿ t � 1 for some natural number t (so thecoefficients of a of the form ai1;...;it are coefficients of a for all �i1; . . . ; it� 2 It) then, forany k 2 N such that 1 % k % t � 1, we define ai1 ;...;i2kn

� ai2knÿk�1;...;i2kn8�i1; . . . ; i2kn� 2 I2kn, and

again ai1 ;...;i2t�2n� a0 8�i1; . . . ; i2t�2n� 2 I2t�2n.

It is clear by the construction of a that it satisfies the properties 1) and 2) mentionedabove.

Now, since M Xÿ1 � �

is an essential R Xh i-submodule of N Xÿ1 � �

, there exists a

polynomial p 2 R Xh i (p � r0 �Psn�1

P�i1 ;...;in�2Yn

ri1;...;in xi1 � � � xin where Yn is a finite subset of In

for all n, 1 % n % s) such that p � a 2M Xÿ1 � �

and p � a �j 0.

For any t 2 N let m�t� � minfn 2 N; nÿ t ^ 2s� 1 and aj1;...;jnÿt ;i1;...;it � ai1 ;...;it

8�j1; . . . ; jnÿt; i1; . . . ; it� 2 Ing (where s is the degree of the polynomial p). We note thatsuch an m�t� exists by 2) above. Let l be any natural number such that 0 % l % s, and let us

prove that� P�i1 ;...;il�2Yl

ri1;...;il

�aj1;...;jt 2M for all �j1; . . . ; jt� 2 It (we set Y0 � f0g).


Since ai1;...;ij � 0 for all �i1; . . . ; ij� 2 Ij and all j such that m�t�=2 < j < m�t� orm�t� < j < 2m�t�, and m�t� ÿm�t�=2 > s, 2m�t� ÿm�t� > s, we see that the term of p � a ofdegree m�t� ÿ l is P

�il�1;...;im�t��2Im�t�ÿl

� P�i1;...;il�2Yl

ri1;...;il � ail ;...;i1 ;il�1;...;im�t� � xÿ1il�1� � � xÿ1

im�t�

�:

Hence, noting that m�t� ÿ t > l, we get that the coefficient corresponding to xÿ1il�1� � � xÿ1

im�t� inp � a, that is,

P�i1;...;il�2Yl

ri1;...;il � ail ;...;i1 ;il�1;...;im�t� �P

�i1;...;il�2Yl

ri1;...;il � aim�t�ÿt�1;...;im�t� , is equal to� P�i1;...;il�2Yl

ri1;...;il

�� aim�t�ÿt�1 ;...;im�t� :

But p � a 2M Xÿ1 � �

means that every coefficient of p � a belongs to M, so we get� P�i1 ;...;il�2Yl

ri1 ;...;il

�aim�t�ÿt�1 ;...;im�t� 2M for all �il�1; . . . ; im�t�� 2 Im�t�ÿl. In other words, we have

just proven that� P�i1 ;...;il�2Yl

ri1 ;...;il

�aj1;...;jt 2M for all �j1; . . . ; jt� 2 It and all l such that

0 % l % s. Of course this holds for any t 2 N:

We now apply the fact that p � a �j 0, which means that there exists some natural l andsome ri1 ;...;il in p such that ri1;...;il � ai1 ;...;in �j 0 for some coefficient ai1 ;...;in of a. Thus, alsori1 ;...;il � a �j 0, and if we let r � P

�i1 ;...;il�2Yl

ri1 ;...;il 2 R we immediately see that r � a �j 0.

Furthermore, r � aj1;...;jm 2M for all �j1; . . . ; jm� 2 Im and all m 2 N, that is,r � a 2M Xÿ1

� �. Hence M Xÿ1

� �% 0N Xÿ1

� �as left R-modules.

We note that the obvious modifications in the proof of i) show that it also holds in the caseof right R-modules.

ii) Suppose M ,! N is an essential monomorphism of left (right) R-modules. IfT � R��RZ R��RZ RR RZ R�� . . ., we know by hypothesis that HomR�T;ÿ�preserves injective envelopes, so HomR�T;M� % 0HomR�T;N� as left (right) T-modules.

Let f be any homomorphism in HomR�RZ R;N�, and let us define f1 � f ,f0 � 02 Hom�R;N� and fn2 HomR�n

R�RZ R�;N� by fn�r1 . . . r2n�� f1�r1 r2 � � � r2n�.Thus we have defined a morphism g � �fi�i ^ 1 2 HomR�T;N�, and we know there exists anelement w 2 T such that wg 2 HomR�T;M�. The element w must be a finite sum of the formw �P

iwi, where each wi is an homogeneous element of degree i, that is, wi 2 i

R�RZ R�for any i, so that wg �P

iwifn� �n ^ 1 (taking note that wifn 2 Hom�nÿi

R �RZ R�;N�, so

wifn � 0 for n % i). We get then that for any n > i,P

iwifn 2 Hom�nÿi

R �RZ R�;M�.Now each wi is also a sum wi �

Pni

k�1kr11 k r12 . . . k ri1 k ri2, so let us consider

the element of R,P

i

Pk

kr11 kr12 � � � kri1 kri2, and the homomorphismPi

Pk

kr11 kr12 � � � kri1 kri2 f 2 HomR�RZ R;N�. For any a b 2 RZ R we have

Pi

Pk

kr11 kr12 � � � kri1 kri2 f� �

�a b� �Pi

Pk

fi�1�a b k r11 . . . k ri2�

��P

iwifi�1

��a b� 2M;


so thatP

i

Pk

kr11 kr12 � � � kri1 kri2 f belongs to HomR�RZ R;M�, and not just to

HomR�RZ R;N�. Of course, it is a nonzero morphism because wg �j 0 (and then wift �j 0for some i and some t > i).

Note that this argument also holds for right R-modules.Conversely, suppose M is an essential left (right) R-submodule of N. Conditions i) and ii)

immediately give that for any index set I, HomR��I�RZ R�;M� is an essential R-submodule of HomR��I�RZ R�;N�; that is, HomR�F;ÿ� preserves injective envelopes ofleft (right) R-modules for any free �R;R�-bimodule F.

Let A be any �R;R�-bimodule, then there exists an epimorphism of �R;R�-bimodulesf : F ! A for some free �R;R�-bimodule F. Since f is a morphism of bimodules, thediagram

HomR�A;M� ÿ! HomR�A;N�?y ?yHomR�F;M� HomR�F;N�

is a pullback diagram of left (right) R-modules. Moreover, HomR�A;M� ! HomR�A;N� isan essential monomorphism of R-modules since HomR�F;M� ! HomR�F;N� is. Therefore,for any �R;R�-bimodule A, the functor HomR�A;ÿ� preserves injective envelopes of left(right) R-modules; hence, if T � R�A��AR A��AR AR A�� . . . , we have thatHomR�T;M� % 0HomR�T;N� as R-modules and then as T-modules. Now chooseA � �I�R�Z R� with I large enough to get the result. h

Proposition 2.3. Let R be any ring (commutative or not). The following statements areequivalent.

i) If I is any index set and fMi; i 2 Ig, fNi; i 2 Ig are two families of left R-modules withMi % 0Ni for all i 2 I, then

Qi2I

Mi % 0Qi2I

Ni.ii) R is left semi-artinian.

P roof. i)) ii) Let T be any left R-module and fTi; i 2 Ig any family of essentialsubmodules. By hypothesis,

Qi2I

Ti is essential inQi2I

T. Consider the homomorphism

f : T ! Qi2I

T given by f �t� � �ti�i2I where ti � t for all i. It is clear then thatTi2I

Ti ÿ! T?y ?y fQi2I

Ti ÿ!Qi2I

T

is a pullback diagram. Hence the inclusionTi2I

Ti ,! T is an essential monomorphism sinceQi2I

Ti ,!Qi2I

T is, and we prove that any intersection of essential submodules is again

essential. Now apply Proposition 1.3 to get that R is semi-artinian.

ii) ) i) Let us prove first that for any family of left R-modules fNi; i 2 Ig we have

soc�Q

i2INi

�%Qi2I

soc�Ni�. For let J be a nonzero simple submodule ofQi2I

Ni; then


J %Qi2I

pi�J�, where pi�J� is the canonical i-th projection of J, and each pi�J� is either simple

in Ni or zero. SoQi2I

pi�J� %Qi2I

soc�Ni� and then J %Qi2I

soc�Ni�. It is immediate to see now

that soc�Q

i2INi

�%Qi2I

soc�Ni�.Now if Mi % 0Ni for any i, soc�Ni� % Mi for any i and then

Qi2I

soc�Ni� %Qi2I

Mi %Qi2I

Ni,

so we get soc�Q

i2INi

�%Qi2I

soc�Ni� %Qi2I

Mi %Qi2I

Ni. It follows thatQi2I

Mi is essential inQi2I

Ni. h

From Theorem 2.2 and Proposition 2.3 we immediately get the following Corollary.

Corollary 2.4. Let R be any ring. The following are equivalent.

i) R is left (right) semi-artinian and the functor HomR�RZ R;ÿ� : R-Mod! R-Mod(Mod-R! Mod-R) preserves injective envelopes.

ii) The functorsQ

: R-Mod! R-Mod (Mod-R! Mod-R) and HomR�RZ R;ÿ� :

R-Mod! R-Mod (Mod-R! Mod-R) both preserve injective envelopes.iii) For any ring homomorphism R! S, the functor HomR�S;ÿ� : R-Mod! S-Mod

(Mod-R! Mod-S) preserves injective envelopes.

Corollary 2.5. Let R be any ring, not necessarily commutative, which contains a field. IfHomZ�R;M� % 0HomZ�R;N� as R-modules whenever M is an essential R-submodule of N,then R is semisimple. In particular, if HomR�S;ÿ� preserves injective envelopes of left (right)R-modules for every R! S ring homomorphism, then R is semisimple.

P roof. If R contains a field then either Z=�p� 7 R for some prime p or Q 7 R, and in anycase, if K is such a subfield (Z=�p� or Q) then for any left (right) R-module M isHomZ�R;M� � HomK�R;M�.

If M % 0N are left (right) R-modules then HomZ�R;M� % 0HomZ�R;N� by hypothesis; butM and N are K-vector spaces, so N �M�L for some vector space L, and then HomZ�R;M�is an R-direct summand of HomZ�R;N�, which means that HomZ�R;L� � HomK�R;L� � 0.This implies L � 0 since R �j 0, so we get M % 0N as left (right) R-modules implies M � N;this is true if and only if R is semisimple. h

Corollary 2.6. Let R be a commutative ring of characteristic n > 0 such thatHomZ�R;M� % 0HomZ�R;N� as R-modules whenever M is an essential R-submodule of N.Then R is artinian.

P roof. If n � pk11 � � � pks

s where pi is a prime number for every i and at least one ki > 0,then R � R1 � . . .� Rs where each Ri has characteristic pki

i . Furthermore, each Ri is suchthat HomZ�Ri;ÿ� preserves injective envelopes, so we can assume the characteristic of R isn � pk for some prime number p and some integer number k > 0. It is then clear thatHomZ�R=pR;ÿ� also preserves injective envelopes, and Z=pZ � R=pR. But Z=pZ is a field,so R=pR is a semisimple ring by Corollary 2.5. This means that R=pR � pkÿ1R is an artinianR-module.

Now consider the exact sequence of R-modules 0! pkÿ1R ,! pkÿ2R!�p pkÿ1R! 0 andget that pkÿ2R is an artinian R-module.


Following this argument we get that pkÿ3R; pkÿ4R; . . . ; p0R � R are artinianR-modules. h

3. Ring extensions and injective covers. We recall from [1] that an injective cover of an R-module M is a linear map � : E!M with E an injective R-module, such that:

i) �� : HomR�E0;E� ! HomR�E0;M� is surjective for any injective module E0, andii) any f : E! E such that � � f � � is an automorphism.

Note that an injective cover of M (if it exists) is unique up to isomorphisms.

Theorem 3.1. If R is a commutative Noetherian ring, then for any ring homomorphismR! S, HomR�S;ÿ� preserves injective covers if and only if R is the product of a finite numberof fields.

P roof. Suppose HomR�S;ÿ� preserves injective covers for any ring homomorphismR! S.

For any prime ideal P of R, let K�P� be the field of fractions of R=P. Then K�P� is anR-module and PK�P� � 0, and if we take any element r 2 R n P, the homomorphism ofR-modules K�P� !�r K�P� is an automorphism. It is easy to argue thatHomR�K�P�;K�J�� 0 for all J 2 Spec �R�, P �j J.

Now given P 2 Spec �R�, let E! K�P� be an R-injective cover. Then for J 2 Spec �R� wehave a ring homomorphism R! K�J�, so HomR�K�J�;E� ! HomR�K�J�;K�P�� is aninjective cover over K�J� by hypothesis.

By the above we then see (noting that K�P�-injective covers are isomorphisms since K�P�is a field)

HomR�K�J�;E� �0 if J �j P

K�P� if J � P :

�So using Matlis� description of injective modules (cf. [2]), we see that E � E�R=P� �E�K�P��. Thus we have that E�K�P�� ! K�P� is a (nonzero) injective cover, and usingMatlis� duality we get K�P�n � K�P� � E�K�P��n � bRP:

We then have depthPbRP � 0, but it is known that this implies depthP RP � 0, which

means that PP 2 Ass�RP�; thus P 2 Ass�R�. Therefore there exists an ideal IP % Risomorphic to R=P. We claim that

PIP is direct.

SupposeP

IP is not direct. Then we can find a minimal number s such thatr1 � � � � � rs � 0 for some distinct primes P1; . . . ;Ps and some 0 �j ri 2 IPi, i � 1; . . . ; s. Wehave then that Pi 7j Pj for some i; j so we can assume P1 7j P2.

Now choose r 2 P1, r2j P2; then rr1 � 0 but rr2 �j 0 so rr2 � rr3 � � � � � rrs � 0 with rr2 �j 0,which contradicts the choice of s.

SinceP

IP is direct with IP �j 0 for all P, and since R is Noetherian, we get that the directsum

PIP is finite; that is, there are only a finite number of IP.

Let P 2 Spec �R�. We know that we have an R-injective cover E�K�P�� ! K�P� (and so itis an RP-injective cover), that K�P� % 0E�K�P�� and that K�P� is a simple RP-module.Moreover, any R-linear K�P� ! E�K�P�� has its image in K�P�, so the kernel ofE�K�P�� ! K�P� is zero and we know then that it is an isomorphism. By Matlis� dualitybRP � HomR�E�K�P��;E�K�P��; and HomR�K�P�;K�P�� K�P� as a ring so bRP � K�P�, a


field. Therefore the Krull dimension of bRP is zero, and this implies that the Krull dimensionof RP is zero; i.e., P is a maximal prime ideal of R.

Since P 2 Spec �R� was arbitrary, we get that R has Krull dimension zero, so each primeideal is maximal. Hence, for M 2 Spec �R�, each K�M� is injective and R has an idealisomorphic to K�M� for each M; thus we easily get that R � K1 � . . .�Ks for a finitenumber of fields K1; . . . Ks.

For the converse, if R � K1 � . . .�Ks then all R-modules are injective. Thus all injectivecovers are isomorphisms and for any ring homomorphism R! S the functor HomR�S;ÿ�preserves injective covers. h

Ac knowl edg e m en ts. We wish to thank Professor Edgar Enochs for his veryinteresting comments and ideas.

References

[1] E. ENOCHS, Injective and flat covers, envelopes and resolvents. Israel J. Math. 39, 189 ± 209 (1981).[2] E. MATLIS, Injective Modules Over Noetherian Rings. Pac. J. Math. 8, 511 ± 528 (1958).[3] A. S. MCKERROW, On The Injective Dimension Of Modules Of Power Series. Quart. J. Math. Oxford

(3) 25, 359 ± 368 (1974).[4] D. G. NORTHCOTT, Injective Envelopes and Inverse Polynomials. J. London Math. Soc. 2(8), 290 ± 296

(1974).[5] S. PARK, Inverse Polynomials and Injective Covers. Comm. Algebra 21, 4599 ± 4613 (1993).[6] S. PARK, The Macaulay-Northcott Functor. Arch. Math. (Basel) 63, 225 ± 230 (1994).[7] B. STENSTRÖM, Rings of Quotients. Berlin-Heidelberg-New York 1975.[8] R. D. WARREN, The Free A-Ring is a Graded A-Ring. Internat. J. Math. Math. Sci. 16(3), 617 ± 620

(1993).[9] T. WÜRFEL, Ring Extensions and Essential Monomorphisms. Proc. Amer. Math. Soc. 69(1), 1 ± 7

(1978).

Eingegangen am 4. 10. 1999

Anschrift der Autoren:

David DempseyDepartment of MathematicsUniversity of KentuckyLexington, KY 40506-0027U.S.A.

Yeong-Moo SongDepartment of Mathematics EducationSunchon National UniversitySunchon 540-742Korea

Luis OyonarteDept. AÂ lgebra y AnaÂlisis MatemaÂticoUniversidad de Almería04120 AlmeríaSpain


A difference matrix construction and a class of balanced generalizedweighing matrices

By

V. C. MAVRON, T. P. MCDONOUGH and C. A. PALLIKAROS

Abstract. In this paper, we obtain new balanced generalized weighing matrices andnew difference matrices. For the former, we use nearfields and adapt a construction ofBerman [1]. For the latter, we generalize a construction of Colbourn and Kreher [5].

0. Introduction. There is a broad family of matrices with subfamilies which are ofimportant independent interest, including difference matrices, generalized Hadamardmatrices, conference matrices and weighing matrices. These matrices are related to groupdivisible designs with a particular type of automorphism group. In some cases, the designs, ortheir duals, are resolvable affine designs and are related, in this way, to resolvable orthogonalarrays. Affine designs, also known as nets, are duals of transversal designs.

In this paper, we construct a family of balanced generalized weighing matrices over newgroups and with new base designs, although their parameters are the same as those ofpreviously constructed balanced generalized weighing matrices. They are constructed usingnearfields. Our method describes the entries of our matrices using simple algebraic relationsin the defining nearfield, and gives a family whose base designs are complements ofMavron�s designs [14]. We note that Berman [1, Theorem 2.2], in constructing families ofgeneralized weighing matrices most of which are not balanced, constructs balancedgeneralized weighing matrices which are the same as ours when the nearfield is a field. Also,de Launey [6] describes a construction for generalized weighing matrices with some of theparameters considered by us. In his construction, however, the groups are cyclic.

We also show that a construction of difference matrices by Colbourn and Kreher [5],defined in terms of some given field, can be generalized in a more combinatorial fashionby the use of other difference matrices. The Colbourn-Kreher construction uses thefield to produce implicitly a conference matrix; we make this part of the constructionexplicit.

Following Cameron et al. [3], we define an orthogonal G-matrix, where G is amultiplicative group, as follows. Let 0 be a symbol not in G and let G � G [ f0g be themultiplicative semigroup in which the product of elements in G is as in the group whileg0 � 0g � 0 for all g 2 G. It will be convenient to adopt the convention 0ÿ1 � 0 in anycalculations below.

Arch. Math. 76 (2001) 259±2640003-889X/01/040259-06 $ 2.70/0� Birkhäuser Verlag, Basel, 2001 Archiv der Mathematik

Mathematics Subject Classification (1991): Primary 05B20; Secondary 05B05, 05B30.

Let C � �ci;j� be a v� b matrix with entries in G and let x and y be integers satisfying1 % x < y % v. Then cx;jcÿ1

y;j , j � 1; . . . ; b, denotes the list of differences between the x-th andthe y-th rows of C. If, for all such pairs �x; y�, the list contains each element of G exactly m

times, where m is a constant, then C is said to be an orthogonal G-matrix of index m. In thiscase, 0 appears bÿ mjGj times in each such difference list. This matrix is referred to as apartial difference matrix by Jungnickel [12], who uses the symbol 1 in place of 0, where heestablishes a relation between such matrices and class regular divisible designs. A differencematrix is an orthogonal G-matrix in which b � mjGj; that is, there are no 0 entries. Ageneralized Hadamard matrix is an orthogonal G-matrix in which v � b � mjGj. Ageneralized conference matrix is an orthogonal G-matrix in which v � b � mjGj � 1. Thisconcept of generalized conference matrix was probably introduced in [12].

If b � mjGj, Jungnickel [11] has shown that b ^ v and that b � v if, and only if, both C andC� � �cÿ1

j;i � are generalized Hadamard matrices. Cameron et al. [3] have shown that ifv � b �j mjGj then C� is also an orthogonal G-matrix.

More generally, an orthogonal G-matrix is said to be a balanced generalized weighingmatrix if there are constants r and k such that

(i) every row has exactly r non-zero entries;(ii) every column has exactly k non-zero entries;

(iii) when each non-zero entry in C is replaced by 1, the resulting matrix is the incidencematrix of a 2-�v; k; l� design where, necessarily, mjGj � l � r�kÿ 1�=�vÿ 1�.

In this case, we say that C is a BGW�v; k; l� over G and we call the design in (iii), the basedesign of the matrix. A 2-�v; k; l� design is said to be signable over the group G if it is thebase design of a BGW�v; k; l� matrix over G.

The base design is trivial in the case of generalized Hadamard matrices and generalizedconference matrices. These matrices are sometimes excluded from the definition of balancedgeneralized weighing matrix by some authors. Moreover, some authors also exclude matricesfor which b �j v, preferring to call these matrices generalized Bhaskar-Rao designs.

We return to the case of an orthogonal G-matrix C of index m and let � : G! H be agroup epimorphism. We extend � in a natural way to a semigroup epimorphism � : G! Hby setting 0� � 0. By applying � to the entries of C, we get an orthogonal H-matrix of indexmjGj=jHj. When C is a balanced generalized weighing matrix, the orthogonal H-matrixobtained by taking H to be the group of order one and � the trivial epimorphism is theincidence matrix of the base design of C.

These matrices have found a variety of applications. For example, Mackenzie and Seberry[13] have used generalized Hadamard matrices to construct certain extremal codes and,more recently, Ionin [10] has used BGW matrices to construct several new families ofsymmetric 2-designs.

In this paper, we construct BGW��qt ÿ 1�=�qÿ 1�; qtÿ1; qtÿ2�qÿ 1�� matrices for anyinteger t ^ 2 over any group G which is the quotient group of the multiplicative group of anynearfield of size q. In particular, q must be a prime power. For odd values of t, the onlypreviously known BGW matrices with these parameters were over cyclic groups. The basedesigns of these new BGW matrices are complements of designs from the family constructedby Mavron [14]. Thus, infinitely many of these complements are signable over certain non-cyclic (indeed, non-abelian) groups.

260 V. C. MAVRON, T. P. MCDONOUGH and C. A. PALLIKAROS ARCH. MATH.

For more details and background of the general topics discussed in this paper see Beth,Jungnickel and Lenz [2] for design theory and difference matrices, Hughes and Piper [9] fordesign theory, The CRC Handbook for Combinatorial Designs [4], Colbourn and Kreher [5]and Geramita and Seberry [7] for difference matrices, and Passman [15] or Huppert andBlackburn [8] for nearfields.

1. Balanced generalized weighing matrices. In this section, we describe a construction forBGW matrices using nearfields. A right nearfield is defined as a set F with two binaryoperations � and � satisfying

(i) �F;�� and �Fnf0g; �� are groups, where 0 denotes the identity element of �F;��;(ii) �x� y� � z � x � z� y � z for all x; y; z 2 F. (The right distributive law).

We denote by F� the multiplicative group �Fnf0g; �� of F. A left nearfield is definedanalogously.

Theorem 1.1. Let F be a finite right nearfield of size q and let t be an integer ^ 2: Thenthere is a BGW��qt ÿ 1�=�qÿ 1�; qtÿ1; qtÿ2�qÿ 1�� over the group F�.

P roof. For each nonzero element of Ft, we will call the first nonzero coordinate itsleading coordinate. Let U denote the subset of elements of Ft with leading coordinate 1. It isimmediate that jUj � �qt ÿ 1�=�qÿ 1�.

Let M be the jUj � jUj-matrix, with rows and columns indexed by the elements of U,whose �x; y�-th entry is defined by Mx;y �

P1 % i % t

xiyi, where x � �x1; . . . ; xt� and y��y1; . . . ; yt�.If each nonzero entry of M is replaced by 1, the resulting matrix is the incidence matrix of

a symmetric 2-��qt ÿ 1�=�qÿ 1�; qtÿ1; qtÿ2�qÿ 1�� design which is the complement of thesymmetric 2-��qt ÿ 1�=�qÿ 1�; �qtÿ1 ÿ 1�=�qÿ 1�; �qtÿ2 ÿ 1�=�qÿ 1�� design F�t� described in[14]. In particular, each row has qtÿ1 nonzero elements.

Now suppose that x0 and x00 are distinct elements of U. Let x0i and x00j be the leadingcoordinates of x0 and x00, respectively, and let a0; a00 2 F. If i < j, there are exactly qtÿ2

elements v 2 Ft withP1 % s % t

x0svs � a0 andP

1 % s % tx00s vs � a00�1�

since the coordinates of v other than vi and vj may be selected arbitrarily. Then, vj isdetermined uniquely from the second equation and vi is determined uniquely from the firstequation. A similar argument applies if i > j.

If i � j, let x000 � x00 ÿ x0. Then, the leading coordinate of x000 is x000l for some l > i. By thepreceding argument, there are exactly qtÿ2 elements v 2 Ft withP

1 % s % tx0svs � a0 and

P1 % s % t

x000s vs � a00 ÿ a0:�2�

But it is clear that every v satisfying (1) also satisfies (2), and conversely.Note also that if v satisfies (1) and if a 2 F then va satisfies similar equations with the

right-hand sides replaced by a0a and a00a, respectively, where va denotes �v1a; . . . ; vta�.If v is nonzero, the set fva : a 2 F�g has exactly one element of U. Hence, the number of

elements v of U for which� P

1 % s % tx0svs

�� P1 % s % t

x00s vs

�ÿ1takes a prescribed value in F� is

qtÿ2. Thus, M is the desired generalized weighing matrix. h

261Vol. 76, 2001 Difference matrices

In the case t � 2 of Theorem 1.1, the BGW matrices are generalized conference matrices.According to [4, Theorem 4.22], the known BGW��qt ÿ 1�=�qÿ 1�, qtÿ1, qtÿ2�qÿ 1��

matrices are over (i) cyclic groups of order dividing qÿ 1 for any integer t ^ 2 or (ii) anygroup of order dividing q� 1 for any even integer t ^ 2, where q is any prime power.

Corollary 1.2. The complement of F�t� is signable over any quotient group of F�, for anyfinite right nearfield F and any integer t ^ 2:

Note that if F is a finite left nearfield, we may proceed as in the theorem above butreplacing the given definition of Mx;y by Mx;y �

P1 % i % t

yixi. In this case, we get a BGW

matrix with the same parameters as above but over the �opposite group� of F�. Here, the basedesign is the dual of the complement of the design F�t� in [14, Theorem 1.1].

Corollary 1.3. Let q be a prime power and let F be any (right or left) nearfield of order q. IfG is any quotient group of F�, there is a BGW��qt ÿ 1�=�qÿ 1�; qtÿ1; qtÿ2�qÿ 1�� matrix overG for all integers t ^ 2:

2. Difference matrices. Let A be a group written multiplicatively and with identity 1. Let 0be a symbol not in A and let A � A [ f0g. We will consider various matrices of the formB � �bi;j�, with entries bi;j 2 A and prove the following theorem.

Theorem 2.1. Let H be a multiplicative group of order n: Suppose that there is a v� w-matrix M � �mi;j�, with entries mi;j 2 bH, satisfying the following conditions:

(i) the number of entries of each row which are different from 0 is a constant k;(ii) there is a constant l such that, in the difference list of two distinct rows, the element 1

appears lÿ 1 times and each h 2 H ÿ f1g appears l times.

Further, suppose that �n� 1��2�k� 1� ÿ �n� 1�l� ^ nw. Then, for any group G of ordern� 1, there is a difference matrix over G with v rows and 2�n� 1��k� 1� ÿ �n� 1�2l columnsand of index 2�k� 1� ÿ �n� 1�l:

Re ma r k 2 . 2 . Suppose that C is a matrix with entries in H satisfying (i) above and

(ii)0 there is a constant l such that, in the difference list of two distinct rows, each h 2 Happears l times.

Suppose also that C has a column with no 0 element. Premultiplying the rows of C bysuitable elements of H, we can arrange that there is a column in which all elements are thesame. Moreover, the resulting matrix still satisfies (i) and (ii)0. Deleting this column leaves amatrix which satisfies (i) and (ii).

Note that any generalized Hadamard matrix or generalized conference matrix (indeed,any orthogonal H-matrix) satisfies (i) and (ii)0.

Re ma r k 2 . 3 . If C is a BGW matrix over H, select a column and delete all rows whichhave a 0 in that column. The resulting matrix satisfies (i) and (ii)0.

The proof of the theorem is through a sequence of lemmas. Let A and A be as definedabove. Define a right action � of A on A by 0�a�� 0 and a0�a�� a0a for all a; a0 2 A. Thus,A has two orbits, a fixed point and a regular orbit, in A.


We induce an action s of A on A� A by �a1; a2��as� � �a1�a��; a2�a�� for all a1; a2 2 Aand all a 2 A. The following lemma is immediate.

Lemma 2.4. The A-orbits in A� A are (i) f�0; 0�g, (ii) f�a; 0� : a 2 Ag, (iii) f�0; a� : a 2 Ag,and (iv) for each a 2 A, f�a0; a00� : a0; a00 2 A; a0a00ÿ1 � ag.

Note that all nontrivial A-orbits in A� A are regular.

Lemma 2.5. Let H and M be as in Theorem 2.1. The v� nw matrix S, with columns indexedby pairs �j; h� where j � 1; . . . ;w and h 2 H, is defined by si;�j;h� � mi;jh. Let 1 % i1; i2 % v andi1 �j i2 and let h1; h2 2 H. Let N�i1 ;i2�h1;h2

be the number of columns �j; h� for which�si1 ;�j;h�; si1 ;�j;h�� h1; h2�. Then

N�i1 ;i2�h1;h2�

n�nlÿ 1ÿ 2k� w� if h1 � h2 � 0;

kÿ nl� 1 if h1 � 0 and h2 2 H or h1 2 H and h2 � 0;

lÿ 1 if h1 � h2 2 H;

l if h1; h2 2 H and h1 �j h2:

8>>><>>>:P r oof. Clearly, if x denotes the number of columns j of M with mi1 ;j � mi2 ;j � 0, then

N�i1;i2�0;0 � nx. Counting the number of columns of M with nonzero entries in both rows i1 andi2, we get wÿ 2k� x � lÿ 1� �nÿ 1�l. Hence, N�i1 ;i2�0;0 � n�nlÿ 1ÿ 2k� w�.

If h1 �j 0 or h2 �j 0, the H-orbit D of �h1; h2� is regular. Let 1 % j % w. Then, �h1; h2� occursonce in the list �si1;�j;h�; si1;�j;h�� where h 2 H if �mi1;j;mi2 ;j� 2 D, and does not occur in this list if�mi1 ;j;mi2 ;j�2j D.

The number of pairs �mi1;j;mi2 ;j� with mi1;j � 0 and mi2;j �j 0 is kÿ nl� 1. Hence,N�i1;i2�0;h � kÿ nl� 1 if h 2 H. Similarly, N�i1 ;i2�h;0 � kÿ nl� 1 if h 2 H.

Now suppose that h1; h2 2 H. The number of pairs �mi1 ;j;mi2;j� in the H-orbit of �h1; h2� iseither lÿ 1 or l according as h1 � h2 or h1 �j h2 by Lemma 2.4. Hence, N�i1;i2�h1 ;h2

� lÿ 1 or l

according as h1 � h2 or h1 �j h2. h

P roof (The ore m 2. 1 ) . Recall that G is a (multiplicative) group of order n� 1. Letf : H ! G be a bijection for which 0f � 1. Form the matrix T from the matrix S in Lemma2.5 by applying f to each entry. Let J be a v� �2�n� 1��k� 1� ÿ nwÿ �n� 1�2l� matrix inwhich every entry is the 1 of G and let D � �di;`� be the v� �2�n� 1��k� 1� ÿ �n� 1�2l�matrix � T J �.

Let g 2 G and let 1 % i1; i2 % v with i1 �j i2. We must determine the number Qg of timesthe pair �ghf; hf�, where h 2 H, occurs as a pair �di1 ;`; di2;`� where ` ranges over all thecolumn indices of D.

If g � 1, there are n�nlÿ 1ÿ 2k� w� pairs corresponding to h � 0 among the columns ofT and 2�n� 1��k� 1� ÿ nwÿ �n� 1�2l pairs among the columns of J. For each h 2 H, thereare lÿ 1 such pairs. Hence, Q1 � n�nlÿ 1ÿ 2k� w� � 2�n� 1��k� 1�ÿ nwÿ �n� 1�2l�n�lÿ 1� � 2�k� 1� ÿ �n� 1�l.

If g �j 1, there are kÿ nl� 1 pairs corresponding to h � 0, kÿ nl� 1 pairs correspondingto h � gfÿ1 and l pairs corresponding to each h 2 H ÿ fgfÿ1g. Hence,Qg � 2�kÿ nl� 1� � �nÿ 1�l � 2�k� 1� ÿ �n� 1�l. Thus, D is the required differencematrix. h

263Vol. 76, 2001 Difference matrices

Theorem 2.1 is a generalization of [5, Theorem 3.2]. In [5], a field is used to constructimplicitly a matrix which plays the role of M in our theorem. It is essentially a generalizedconference matrix adapted as described in Remark 2.2.

References

[1] G. BERMAN, Families of generalized weighing matrices. Canad. J. Math. 30, 1016 ± 1028 (1978).[2] T. BETH, D. JUNGNICKEL and H. LENZ, Design Theory, 2nd edition. Cambridge 1999.[3] P. J. CAMERON, P. DELSARTE and J.-M. GOETHALS, Hemisystems, orthogonal configurations and

dissipative conference matrices. Philips J. Res. 34, 147 ± 162 (1979).[4] C. J. COLBOURN and J. H. DINITZ (eds.), The CRC Handbook of Combinatorial Designs. Boca

Raton 1996.[5] C. J. COLBOURN and D. L. KREHER, Concerning difference matrices. Designs, Codes and

Cryptography 9, 61 ± 70 (1996).[6] W. DE LAUNEY, GRBDS ± Some new constructions for difference matrices, generalized Hadamard-

matrices and balanced generalized weighing matrices. Graphs and Combin. 5, 125 ± 135 (1989).[7] A. V. GERAMITA and J. SEBERRY, Orthogonal Block Designs. New York 1979.[8] B. HUPPERT and N. BLACKBURN, Finite Groups III. Berlin-Heidelberg-New York 1982.[9] D. R. HUGHES and F. C. PIPER, Design Theory. Cambridge 1985.

[10] Y. J. IONIN, Building symmetric designs with building sets. Designs, Codes and Cryptography 17,159 ± 175 (1999).

[11] D. JUNGNICKEL, On difference matrices, resolvable transversal designs and generalized Hadamardmatrices. Math. Z. 167, 49 ± 60 (1979).

[12] D. JUNGNICKEL, On automorphism groups of divisible designs. Canad. J. Math. 34, 257 ± 297 (1982).[13] C. MACKENZIE and J. SEBERRY, Maximal q-ary codes and Plotkin�s bound. Ars Combin. 26B, 37 ± 50

(1988).[14] V. C. MAVRON, A class of symmetric designs. Geom. Dedicata 1, 468 ± 476 (1983).[15] D. S. PASSMAN, Permutation Groups. New York-Amsterdam 1968.

Eingegangen am 22. 9. 1999

Anschrift der Autoren:

V. C. Mavron, T. P. McDonoughDepartment of MathematicsUniversity of WalesAberystwyth, SY23 3BZUnited Kingdom

C. A. PallikarosDepartment of Mathematics and StatisticsUniversity of CyprusP.O. Box 5371678 NicosiaCyprus


ring extensions, injective covers and envelopes

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