Math. Z. 110, 15-26 (1969)

Structure of 1-Primitive Near-Rings D. RAMAKOTAIAH

Introduction

In ring theory there is the following density theorem for irreducible mod- ules (topological formulation [4, p. 31]). Let U be an irreducible ring of endo- morphisms acting on the commutative group (M, + ) and let F be the centralizer of M. Then U is (topologically) dense in the ring L of linear transformations of M over F into itself.

The object of this paper is to prove a comparable result for near-rings and to determine the class of all 1-primitive near-rings with unity element. For this purpose, we have introduced the notions of a centralizer of an arbitrary group (see Def. 2) and the N-centralizer of a N-group (see Def. 4). In case M is a N-group of type 2, the N-centralizer of M is a centralizer of the group M (Theorem 5).

This paper has been divided into two articles. In w 1, a density theorem for N-groups of type 1 with A(M)=(0) is proved (Theorem 4). In w 2, it is proved that every 1-primitive near-ring N (which is not a ring) with unity element is dense in the near-ring of all transformations of M over F (see Def. 3) where M is a N-group of type 1 with A(M)= (0) and F is the N-centralizer of M (Theo- rem 7). Further it is proved that, for each centralizer S of a group M, every dense subnear-ring of the near-ring of all transformations of M over S is a 2-primitive near-ring (Theorem 8).

The referee has brought to my notice that the basic result of this paper, Theorem 4, was known to Wielandt as early as 1937; although he presented the proofs at various meetings only a short summary without indication of proofs appeared in print [6].

Preliminaries. We recall (see Betsch [2]) that a (right) near-ring N = (N, + , .) is a system where i) (N, + ) is a (not necessarily abelian) group which we denote by N +, ii) (N, .) is a semigroup, iii) a(b+c)=a b+a c for all a, b, c in N and iv) 0 a = 0 for all a~N, where 0 is the identity element of N +. N-groups (N-spaces in the terminology of [3]), right ideals and ideals of a near-ring N, etc., are defined in the usual way [1 - 3].

A N-group M+(0) is called irreducible if and only if it has no proper N-subgroup (+0) which is the kernel of some N-homomorphism of M; and a N-group M is called cyclic if and only if for some m~M, m N = M and any such m is called a generator of M. An irreducible N-group M is said to be of

type 0, if and only if M is cyclic,

type 1, if and only if M is cyclic and for each meM, either raN=(0) or m N = M ,

type 2, if and only if M (is cyclic and) has no proper N-subgroups (=t= 0).

16 D. Ramakomiah :

Following Betsch [1], we will call a near-ring v-primitive if and only if there is a N-group M of type v with A(M)=(0), where A(M) is the set of att annihilators of M in N. We observe that a 2-primitive near-ring is a 1-primitive near-ring and hence any result which is valid for I-primitive near-rings is also valid for 2-primitive near-rings. Further for a near-ring with unity element 1-primitivity and 2-primitivity are equivalent, since the unity element of N acts on every cyclic N-group as the identity mapping.

Now suppose M is a N-group and w~M; then the set A(w) of all the anni- hilators of w in N, that is, {x ~ N/w x = 0} is a right ideal of N and if L is a right ideal o f N, then w L= {ws/seL} is a N-subgroup of M. Finally we remark that a nonempty subset M~ of a N-group M is the kernel of some N-homo- morphism of M if and only if i) (M1, +) is a normal subgroup of M and ii) (m+w) x - w x E M 1 for all m~M1, w~M and x~N. [1,4.1, [1]]. We now prove

Lemma 1. If M is a N-group with a generator w and L a right ideal of N, then w L is the kernel of a N-homomorphism of M.

Proof Clearly wL is a normal subgroup of M. Let wleM; since w is a generator of M, wt=w~ for some e~N. Now for each w[J~wL, xEN, we have ( w l ~ + w O x - w l x = w { ( ~ + 7 ) x - c ~ x } e w L and hence wL is the kernel of a N-homomorphism of M.

Corollary 1. If M is a N-group of type 1 and L a right ideal o[" N, then for each m~M, either mL= (0) or M.

Pro@ tf mL+(O), then mN=f=O) and hence m N = M . Therefore, m is a generator of M. Since M is irreducible and since m L #:(0) is the kernel of a N-homomorphism of M, we have m L = M.

Corollary 2. I f M is a N-group of type I with A (M) = (0) and L + (0) a right ideal qf N, then there exists an element m e M such that m L = M.

Proof Since A(M)=(0), there exists an element m~M such that mL+(O). Now by the previous corollary, it follows that mL=M.

w

Let M be a N-group, We denote the set {w~M/wN=(O)} by C O and the complement of C o in M by M/C o . I fM is a N-group of type 1 and i fN contains a unity element, then C o = (0), where 0 is the identity element of M.

Theorem 1, Let M be a N-group of type 1. I f m~M/Co, then A(m) is a maximal right ideal of N and the quotient group N + - A(m) is N-isomorphie to M.

Proof Since mEM/Co, mN+(O) and hence raN=M, Consequently the mapping qS :x~mx is a N-homomorphism of N + onto M, Since A(m) is the kernel of qS, A (m) is a right ideal of N and N + -A(m) is N-isomorphic to M, Since M is a N-group of type 1, it now follows that A(m) is a maximal right ideal.

Corollary 3. If w, w ~ l / C o and if A(w)~_A(wl) , then A(w)=A(w 0

Structure of l-Primitive Near-Rings 17

We now prove the following impor t an t

Theorem 2. Let N be a 1-primitive near-ring which is not a ring and M a N-group of type 1 with A(M)=(0) . Let w, Wl, ..., w, e M / C o . I f

A(w)~_ (~ A(wi) , i = 1

then A (w)= A(wi) for at-least one i, 1 <_i<_ n.

The p roo f of this theorem requires a couple of l emmas

L e m m a 2. Let M be a N-group with A(M)=(0) . I f N + is noncommutative, then M is also noncommutative.

Proof. Since N + is not commuta t ive , there exist ct, f i eN + such that ct+fl ~ f l + a . Since A(M)=(0 ) , there exists an m e M such that m(e+fi -e- f l )~=O, tha t is, m c~ + m fi + m fi + m ~ and hence M is not commuta t ive .

L e m m a 3. Let M be a N-group with A(M)=(0) . I f there exist ~, fi, S e N such that (~ + fl) g) =4= c~ 5 + fi 5, then there exist ml, m 2 ~M such that

(ml+ m2) 54= rn 1 5 + m 25.

Proof Let c~, fi, 6 e N such that (e + fi) 6 + c~ (5 + fl 5. Since A (M) = (0), there exists an element m e M such that m { ( ~ + f i ) 5 - f i 6 - ~ 5} =t=0, that is,

(m~ +m fi) 5 * (mc~) 6 + (m fi) 5.

N o w take m l = m a and m2=m ft.

Proof of Theorem 2. By the Corol la ry 3, it follows that the theorem is true for n = 1. Assume the t ruth of the result for n. Suppose w, % , . . . , w,§ are elements of M/C o such that

n + l

A(w)=_ A(wO. i = 1

Set n

R = ~ A(wi). i = 1

Suppose A (w) ~ R, A (w) ~ A (w, § i). Therefore, w R 4= (0), wA (w, + 1) ~ (0). Since R and A(w, +1) are right ideals of N, we have w R = M = w A(w, + 1) (Corol lary 1).

Case 1. Suppose N + is not commuta t ive . Let mx,m2eM. There exist Yl e R, Y2 cA (w, + 1) such that w Yl = mi for i = 1, 2. Since R and A (w, + 1) are no rma l subgroups of N + , y l + ( y 2 - Y l - Y 2 ) e R and ( y l + y 2 - y l ) - y 2 e A ( w , + l ) and hence y l + Y 2 - y l - y 2 e R c ~ A ( W , + l ) . Therefore, w ( y l + y 2 - Y l - Y 2 ) = O , that is, w Yl -I- w Y2 --= w Y2 d- w Yl and hence m 1 + m 2 =- m 2 d- m 1 . Therefore, M is c o m m u - tative. Since N § is not commuta t ive , by L e m m a 2, it follows that M is not commuta t ive , a contradict ion. Therefore, if N + is not commuta t ive , then A(w) must contain atleast one of R and A(w,+l). 2 Math. Z., Bd. I10

18 D. Ramakotaiah:

Case 2. Suppose N + is commuta t ive . Since N is not a ring, there exist e, fl, 8 e N such that (c~ + fl) 8 + c~ 8 + fl 8. N o w by L e m m a 3, there exist m~, m 2 e M such that (m 1 + m2) 8 @ m 18 + m 28. Since w R = M = wA(%+l) , there exist yl~R, y2eA(w,+l) such that wyi=m i for i = 1, 2. Since y l e R and since R is a right ideal of N, ( y l+y2)8=y28+r l for some r l~R. Since Y2eA(%+l) and since A(w,+l) is a right ideal, (Y2 + Yl)8 = Yl 6 + r 2 for some r 2 eA(W,+l). N o w

Therefore,

Y2 8 "~-/'i = (Yl -~- Y2) 8 = (Y2-t-Yl) 8 = y l 8 -}- r2"

Yl 8 -- r 1 = Y 2 8 - - r 2 = ~ e R c~A(w,+x)~_A(w ).

Therefore, (w Y0 8 = w r 1 and (w Y2) 8 = W r 2 . N o w

(m 1 -t- m2) 8 = (w Yt + w Y2) 8 = w (Yl + Y2) 8 = w (Ya 8 + r2) ---- (w Yl) 8 + w r 2

= ( w y 0 8+(w y2) 8=ml S+m 2 8,

a contradict ion. Therefore, in this case also A(w) must contain atleast one of R and A(w,+ O.

N o w if A(w)_ A ( % + 0, by Corol la ry 3, we have A(w) = A(w, + 1). If A (w) ~_ R, then by our a s sumpt ion A(w)=A(wi) for atleast one i , l <_iNn. In either case, we have A(w)= A(w~) for atleast one i, 1 < i < n + 1. Hence the result.

In the remaining par t of this article we assume that N is a 1-primitive near-r ing which is not a ring and M a N - g r o u p of type 1 with A(M) --- (0).

Theorem 3. For each n> 1, let S ,= {wt, ..., w,} be a subset of M/C o such that A(wi)=~ A(wj) for i@j. Then N + - A ( S , ) is a direct sum of n copies of M.

Proof The p roof is by induction. The truth of the result for n = 1 would follow f rom T h e o r e m 1. Assume the t ruth of the result for m. Let Sin+l= {wl, . . . , w,,+l } be a subset of M / C o such that A(wi)@A(w~) for i@j. N o w by the T h e o r e m 2, A(wm+O~A(Sm). Since A(Wm+l) is a max imal right ideal, we have A(wm+O+A(Sm)=N. N o w proceeding as in the T h e o r e m 3.3, [-5], the result would follow.

Lax ton has ob ta ined this result [see 3.3, [-5]] for pr imit ive distr ibutively generated near-rings with unity element. Therefore, the above theorem is a general izat ion of Lax ton ' s result [3.3, [5]] .

It is now convenient to in t roduce the following

Definition 1. Let M be a N-group and Wa, W E G M . Then w I is equivalent to w 2 - in symbols w 1 ~ w E - if and only if A(w 0 = A(w2).

Clearly ~ is an equivalence relat ion; and the equivalence class containing zero is precisely C 0, the set of all elements of M which are annihi lated by N.

Corol lary 4. I f N satisfies descending chain condition (d. c. c.) for right ideals, then M has only a finite number of equivalence classes.

Structure of 1-Primitive Near-Rings 19

Proof, If the number of (distinct) equivalence classes of M under ,-~ is not finite, then we can pick a countable family {C.} of distinct ~ equivalence classes such that each C, 4 = C o . Now let w,e C, and put

n

R,= (~ A(wi). i = l

Now {R,} is a descending chain of right ideals of N and hence there is an integer m such that R , = R m for all n>m. Since, for n>m, A(w,)~_R,=Rm, it tbllows by Theorem 2, A(w,)=A(wi) for atleast one i, l<i<_m. Hence the set {C,}= {Ca, C2, .,., C,,}, which is a contradiction. Therefore, M has only a finite number of equivalence classes.

Corollary 5. I f M =g O) has only a finite number of ~ equivalence classes, then there exist wt, w2, . . . , W a of M such that

1 ~) A (w~) = (0). i = a

Proof Let Co, Ct, . . . , C, be the distinct equivaIence classes of M under the equivalence relation ~ and let w~ be a representative element of C~ for i = 1 . . . . . n. If

(~ A (w,) + (0), i = l

by the Corollary 2, there exists a w r such that

w 0 A(w,) = M i = a

and consequently it follows that A(w)~ A(wi) for i= 1, ..,, n and hence w does not belong to any one of the equivalence classes Co, Ca, ..., C, whose union is M, a contradition. Therefore,

0 (0). i = 1

If S is a subset of N, we denote the set {w ~ M/w S = (0)} by S • Clearly Co, the set of all elements of M which are annihilated by N, is a subset of S j-. We denote the complement of C O in S l by S•

Lemma 4. Let w 1 . . . . , w, be elements of M / C o belonging to the equivalence classes Ct, C 2 . . . . , C,. Then

f ~ \ • /Co

i = l

Proof Suppose w~ C~ for some i. Then wA(w~)=(O) and hence

W W i ~ .

\ i = 1 /

2*

20 D. Ramakota iah:

Therefore,

Since we Ci, we Co. Therefore,

we(i~= lA(Wl)) l / C o �9

Suppose / n , , 2 / / n , ,•

w e ( ~ A(wi) ) / C o, then w e ( ~ A(w~)) x i ~ 1 / ~ \ i = 1 /

Since

and w~ C o.

we A(wi , w A(w~)=(O). \ i = 1 / i = 1

N e M / C o , N r andhence N r A(wi) �9 \ i = 1

i * j Therefore,

wj( ~ A(w'i))'-M i = 1 ] i * j

(Corollary 1). Hence there exists an element n

ej e 0 A (wi) i = 1 i , j

i,j Since

Therefore, n

A(w)~_ ~ A(wi), i = 1

By the Theorem 2, it follows that A(w)= A(w~) for some i and hence we C i_~ C 1 u . . -w C,.

We now prove the following density theorem for N-groups.

Theorem 4. (First density theorem.) Let N be a 1-primitive nearring which is not a ring and M a N-group of type 1 with A(M)=(0). Let wl, w2, ... , w, be elements of M/C o belonging to n different equivalence classes C1,..., C, respec- tively, l f m 1 .. . . , m, are any n elements of M, there exists an element beN such that wib=mi for i= l, .. , n.

Proof For each j, set ~ = C 1 ~ C 2 •...w Cj_ 1 w Cj+ 1 w.. .w C,, then wjr Tj. By the previous lemma,

Structure of l -Primit ive Near -Rings 21

such that wj ej = m~ and we have wi ej = 0 for i+j. Writing

b= ~ ej, j = l

we have w~ b = m~ for i = 1, 2, ..., n.

Corollary 6. For each n> 1, let S, = {wl, .,., w,} be a subset of M / C o such that A(wi)#: A(wj) for i#:j. Then there exists an element e ~ N such that w~ e = w i for i= 1, 2 , . . . , n and

n

a - e a e O A ( w i ) foral l a e g . i = l

Proof By the density theorem, there exists an e 6 N such that w i e=w~ for i = l , . . . , n . Now for each e s N , wiec~=wie and hence w~(ec~-e)=0 for i= 1, 2, ..., n. Therefore,

n

7 - e ~ (~ A(w~). i=1

Corollary 7. Every 1-primitive near-ring satisfying d.c.c, for right ideals has a left identity.

This follows from the corollaries 4, 5 and 6 or from [1 ; 3.4a].

w

In the present article we obtain a second density theorem for a 1-primitive near-ring with unity element (near-rings with this property are 2-primitive) and derive the class of all 1-primitive near-rings with unity element. However, it is possible to obtain a few results which are more general than those required for this purpose which seem interesting in themselves.

Suppose M is an arbitrary group and ~4# the set of all mappings of M into itself. Now JV satisfies the first three conditions of a near-ring under the usual addition and the iteration. We now introduce a topology in Y by taking the sets As(w ) = {r~J1/"/w T= w f } (where w ~ M and f~ J~) as a subbasis. It can be verified easily that every neighbourhood o f f e ~ contains an open set of the form

f + (~ A0 (wl), i = I

where w i ~ M and 0 is the zero mapping of M into itself.

It is now convenient to introduce the following

Definition 2. Let M be an arbitrary group. S is called a centralizer of M if and only if

i) the zero endomorphism 0 of M belongs to S;

ii) S'\0 (the complement of O in S) is a group of automorphisms of M;

iii) 49, ~ ~ S and w 49 = w ~ for some nonzero element w of m imply 49 = ~.

22 D. Ramakotaiah:

Definition 3. Let M be an arbitrary group and S a centralizer of M. Then a mapping of M into itself is called a transformation of M over S if and only if it commutes with each element of S.

If 0 M is the identity element of M and T a transformation of M over S, then 0g T=(0M0 ) T=(0 M T)0=0 M. Thus each transformation of M over S fixes the identity element of M and we denote the set of all transformations of M over S by N(S). Now N(S) is a near-ring under the usual addition and the iteration and M can be considered as a N(S)-group, If S consists of 0 and the identity mapping e of M alone, then N(S) is the near of all mappings of M into itself which fix the identity element of M.

Lemma 5. I f S is a centralizer of M, then N(S) is a closed subset of ~A r.

Proof Let N(S) be the closure of N(S), fEN(S) and CES. For each wEM, f+Ao(w)c~Ao(w4 ) intersects N(S). Therefore, there exists a TEN(S)c~ ( f+ A o (w) n A 0 (w 4)) and hence w T= wf and (w 4) T= (w 4 ) f Since TeN(S), T commutes with each element of S and hence (w 4) f = (w 4) T= (w T) 4 -- (w f ) 4 . Therefore, f commutes with each element of S and hence feN(S), that is, N(S) c N(S). Therefore, N(S) = N(S).

In particular the near-ring of all mappings of M into itself which fix the identity element of M is a closed subset of JV'.

We are now in a position to obtain the second density theorem. We begin with the following

Definition 4. Let N be an arbitrary near-ring and M a N-group. The set of all N-endomorphisms of M, that is, those endomorphisms 4 of M for which (wx)4=(w4)x , for all xEN, is called the N-centralizer of M and it will be denoted by F.

Lemma 6. Any nonzero N-endomorphism of a N-group M of type 1 is an automorphism of M.

Proof (The following proof has been suggested by the referee.) Let 4 be a nonzero N-endomorphism of M. Since M is a N-group of type 1, the kernel of 4 must be either M or (0). Since 4 :# 0, it follows that the kernel of 4 is (0) and hence 4 is an injection. Now let m be a generator of M. If m 4 ~ Co, then (0)--- (m 4) N = (m N) 4 --- (M) 4 and hence 4 = 0, a contradiction. Therefore, m4EM/C o and hence M = (m 4) N = (m N) 4 = (M) 4. .Therefore, 4 is a surjec- tion. Hence 4 is an automorphism of M:

Theorem 5. The N-centralizer F of a N-group M of type 1 consists of O and a group of automorphisms of M. I f M is a N-group of type 2, then the N-central- izer F of M is the centralizer of the group M.

Proof Clearly 0EF. By the Lemma 6, it follows that each nonzero element ofF is an automorphism of M. It can be verified that the nonzero elements o f f form a group. Now let M be a N-group of type 2. Suppose there is a nonzero element w E M such that w 4--w ~p for some r ~ e F. Let P = {w E M/w 4 = w ~}. It can be verified that P is a N-subgroup of M. Since M is a N-group of type 2

Structure of 1-Primitive Near-Rings 23

and since P+(0) , it follows that P=-M and hence w4)=wtp for all weM(=P). Therefore, 4) = ~ and hence F is a centralizer of the group M.

T h e o r e m 6. Let N be a near-ring with unity element, M a N-group of type 1 and wl, w2~M. Then w l ~ w 2 if and only if there exists a 4)eF/O such that w 14) = w2, where F is the N-centralizer of M.

Proof. Suppose w14)=w 2 for some 4)eF/O. Since 4)+0, 4) is an auto- morphism of M (Lemma 6). Since F/O is a group of automorphisms of M (Theorem 5), 4)-1~F/0 and w 2 4)-1 = wl" Now, ifxeA(wO, then w 2 x =(w 14)) x = (w 1 x) 4)= 0 so that x E A(w2), Therefore, A(w2)~_ A(wl). If x eA(w2), then wt x = (w2 4)-1)x=(w2x ) 4 ) -1=0 so that xeA(w 0 and hence A(wO~_A(w2). There- fore, A(wl)= A(w2) and hence w 1 ~ w 2 . Conversely, suppose that wl ~ w2. Since C o = (0), it follows that either w 1 = w2 = 0 or w 1 + 0, w2 :I: 0. If wl = w2 = 0, then for any 4)eF/O, w14)=wz. Suppose w~ :#0, w 2 , 0 ; since N has a unity element w 1 N = M = w 2 N. Consider the map 4): M ~ M given by (w ix) 4)=w 2 x. If w 1 x = 0, then x eA(w 0 = A(w2) and hence w 2 x = 0. Therefore, 4) is well defined. It can be verified that 4) is an N-endomorphism of M. Since N contains a unity element, we have w 14)= w 2 and since 4 ) , 0, 4)~ F/O (Lemma 6).

We assume that N is a 1-primitive near-ring with unity element and M a N-group of type 1 with A(M)=(0). Since N contains a unity element, M is a N-group of type 2. Therefore, the N-centralizer F of M is a centralizer of the group M (Theorem 5). Every right multiplication of elements of M with elements of N is a transformation of M over F and hence N determines a near-ring of transformations of M over F. Since A (M) = 0), N can be considered as a subnear-ring of N(F). M can be considered as a N(F)-group of type 2 and the N(F)-centralizer of M is F itself. We now introduce a relation ~ ' in M as a N(F)-group as follows: If wl, wz~M, then wl,,Jw 2 if and only if A ' ( w 0 = A' (wz), where A' (w)= { TsN(F)/w T= 0}. Clearly ,--' is an equivalence relation. Thus we have two equivalence relations ~ and --J on M as a N-group and a N(F)-group respectively. We now prove the following

L e m m a 7. ,-~ = ~'.

Proof. If w 1 ~ ' w2, then A' (wl) = A' (wz) and hence N c~ A' (wa) = N ca A' (w2), that is, A(wO=A(wz). Therefore, wa~w 2. Conversely suppose that wl,,~w 2. Now there exists a 4)sF/O such that wl 4)=w 2. Now N(F) contains a unity element and M is a N(F)-group of type 1. Moreover the N(F)-centralizer of M is F itself and wl 4)= w2 for some 4)~F/O. Therefore, w 1 ~ ' w 2 (Theorem 6).

We come now to the topological formulation of the density theorem.

T h e o r e m 7 . (Second density theorem.) Let N be a 1-primitive near-ring (which is not a ring) with unity element and M a N-group of type 1 with A (M) = (0). Then the closure of N in ~ is N(F), where F is the N-centralizer of M.

Proof. Let f eN(F) and

n

P = f + (']A'(wi), where A'(wl)=Ao(wi)c~N(F ). i = 1

24 D. R a m a k o t a i a h :

We may assume that w~, ..., w, are nonzero and nonequivalent under the equivalence relation ~'. By the previous lemma, ~ = --~'. Therefore, wl, w 2 .. . . , w, are nonequivalent under the equivalence relation ~ and belong to M / C o = (0). Now by the first density theorem there exists an element b e N such that w i b = w i f for i = 1, ..., n. Consequently

b e f + (~ A' (wl). i=1

Therefore, P meets N. Since any neighbourhood of f contains an open set of the form P, it follows that each neighbourhood of f meets N and hence N is dense in N(F). Therefore, N ( F ) c N ; since N o N ( F ) and N(F) is closed, it follows that the closure of N in A r is N(F).

Corollary 8. Let N be a 1-primitive near-ring (which is not a ring) with unity clement satisfying d.c.c, for right ideals and M a N-group of type 1 with A(M) = (0). Then N = N(F), where F is the N-centralizer of M.

Proof Since N satisfies d. c. c. for right ideals, M has only a finite number of -equivalence classes (Corollary 4). But ~ = ~'. Therefore, M has only a finite

number of ~'-equivalence classes. Therefore, there exist Wl, ..., w, e M such that

A ' (w~) = (0) i = i

(Corollary 5).

Since N is dense in N(F), it follows that N=N(F) .

We now show that for each centralizer S of a group M, each dense subnear- ring of N(S) (the near-ring of all transformations of M over S) is a 2-primitive near-ring. For this we introduce the following

Definition 5. Let M be an arbitrary group and S a centralizer of M. Then wl, w z e M are said to be S-equivalent - in symbols Wl~W 2 - if and only if there exists a r e S/O such that w I 0 = wz.

Clearly ~ is an equivalence relation and if C is an equivalence class not containing zero and w a fixed element of C, then C= {w 05/05eS/0} and the w r are distinct (Definition 2, iii)).

Lemma 8. Let M be a group, S a centralizer of M and w 1 q= O, w 2 elements of M. Then there exists a TeN(S) such that w I T=w 2.

Proof Suppose w 1 belongs to the equivalence class C under the equivalence relation ,.~. Define a map T: M - ~ M as follows: Let meM. Ifm~ C, put m T=0. If meC, there exists a unique CeS/O such that m=w~05. Now put roT= W10 5 T = w 2 05. Clearly T is well defined and w, T = W 2. Now let m e M , 05~S. If 05 = 0, then (m 05) T= (m T) 05 = 0. Suppose 05 = 0. If m r C, then m 05 q! C and hence m T = (m 05) T = 0. Therefore, (m T)05=(m 05)T. If me C, there exists a unique OeS/O such that m=w,O. Now (m05) T-(wlO05) T=(w2O)05= (wa ~) T05 = m T05. This shows that T commutes with S and hence belongs to N(S). Hence the result.

Structure of t-Primitive Near-Rings 25

Corollary 9. Let M be a group, S a centralizer of M and N a dense subnear- ring of N(S). I f w 1 ~=0, w 2 are elements of M then there exists a TeN such that w 1 T= W 2 .

Proof Now by Lemma 8, there exists an f eN(S) such that w~ f = w 2 . Since N is dense in N(S),f+(Ao(wl)c~N(S)) meets N and hence there exists a TeN such that w 1 T= w 1 f. Therefore, w 1 T= w 2 .

Theorem 8. Let M +(O) be a group and S a centralizer of M. Ther~ every dense subnear-ring N of N(S) is a 2-primitive near-ring.

Proof Now M can be considered as a N-group. By the Coroliary 9, it follows that M has no proper N-subgroups (4: 0) and each nonzero element of M is a generator of M. It can be verified that A(M), the set of all annihilators of M in N, is (0). Therefore, M is a N-group of type 2 with A(M)= (0) and hence N is a 2-primitive near-ring.

In particular any dense subnear-ring of N(S) with unity element is a 2-primi- tive near-ring with unity element and hence a 1-primitive near-ring with unity element. Thus we have determined the class of all 1-primitive near-rings with unity element (Theorems 7 and 8).

We give an example showing that Theorem 7 need not be true for 1-primi- tive near-rings without unity element. For this, we consider the following exampte given by Betsch. Let G = (0, 1, 2, 3) be the group of integers rood 4 and A = (0, 2). Let N be the set of all mappings of G into itself which leave the zero fixed and map d onto 0. Now N is a near-ring and it can be verified that G is a N-group of type 1 with A(G)=(0). Therefore, N is a l-primitive near-ring and N has no unity element. The only automorphisms of G are the identity mapping and the mapping q5 given by 0 ~b = 0, 2 ~ = 2, 1 q5 = 3, 3 ~b = 1. The map T given by, 17"-- 3 and 0 T= 2 T - 3 T= 0 belongs to N and does not commute with ~b. Consequently the N-centralizer F of G consists of {0, e} and hence N(F) is the near-ring of all mappings of G into itself which fix 0. Now

A(1) c, A(2~ n A(3)= (o),

where A(r)= {TeN(F)/rT-O} for r = 1, 2, 3. If N is dense in N(F), it follows that N = N(F), which is not the case. Therefore, N is not dense in N(F).

We give another example showing that the Theorem 7, cannot be extended for 0-primitive near-rings with unity element. Let N be the set of all mappings of G into itself which fix zero and which map A into itself. Now N is a near-ring and G is a N-group of type 0 with A(G)=(0). Therefore, N is a 0-primitive near-ring and N has a unity element. Since 2 N + G, 2 N + 0 , G is not a N..group of type 1. Now proceeding as in the above example, it follows that N is not dense in N(F). Further, there is no TEN such that 2 T= 1 and hence Theorem 4 cannot be extended to N-groups M of type 0 with A(M)= (0).

We recall that two sets S 1 and S 2 are said to be equipotent if and only if there exists a bijection of S 1 onto S 2 .

26 D. Ramakotaiah: Structure of 1-Primitive Near-Rings

Theorem 9. Let N be a near-ring with unity element and M a N-group of type 1. Then any two equivalence classes other than C o (under the equivalence relation ~ ) are equipotent.

Proof Since N contains a unity element and since M is a N-g roup of type 1, it follows that M is a N-g roup of type 2. Let C be an equivalence class other than Co and w a fixed element of C. Consider the map f : q5 ~ w 4), for 43sF/O, where F is the N-centralizer of M. By the Theorem 5, it follows that f is an injection of F/O into C and by the Theorem 6, it follows that f is a surjection. Therefore, f is a bijection and hence any two equivalence classes other than Co are equipotent.

Theorem 10. Let N be a finite 1-primitive near-ring with unity element (which is not a ring) and M a N-group of type 1 with A(M)= (0) and with order n, where n - 1 is a prime. Then either N is isomorphic with the near-ring of all mappings of M into itself which f ix zero or N + is N-isomorphic to M.

Proof Let r be the number of equivalence classes other than C o . Since the equivalence classes o ther than Co are equipotent, it follows that r / n - 1. Since n - 1 is a pr ime we have either r = n - 1 or r = 1. If r = n - 1, then each equiv- alence class contains just one element and hence F/O contains only the identity element. Therefore, F = ( 0 , e) and hence N(F) is the near-ring of all mappings of M into itself which fix zero. By the Corol lary 8, it follows that N = N(F), that is, N is the near-r ing of all mappings of M into itself which fix zero. If r = 1, then there is only one equivalence class other than C o. Since A(M)=(0) , it follows that N + ~ M (Theorem 3).

I take this opportunity to thank the referee and Dr. N. V. Subrahmanyam for their valuable suggestions and criticisms.

References

1. Betsch, G.: Strukturs~tze fiir Fastringe. Inaugural-Dissertation, Universit~it Tiibingen 1963. 2. - Ein Radikal ffir Fastringe. Math. Z. 78, 86-90 (1962). 3. Blackett, D. W.: Simple and semi-simple near-rings. Proc. Am. Math. Soc. 4, 772-785 (1953). 4. Jacobson, N.: Structure of rings. 1. Edition. Am. Math. Soc. Coll. Publ. Vol. 37. Providence

R.I., 1956. 5. Laxton, R. R,: Primitive distributively generated near-rings. Mathematika 8, 142-158 (1961). 6. Wielandt, H.: ~ber Bereiche aus Gruppenabbildungen. Deutsche Math. 3, 10 (1938).

Dr. D. Ramakotaiah Post-Graduate Center Andhra University Nattapadu, Guntur-5 (India)

(Received April 2, 1966)