Acta Math. Hungar.

89 (3) (2000), 205{210.

THE NIL RADICALS IN SEVERAL CLASSES

OF LATTICE-ORDERED RINGS

GAO TING� (Shijiazhuang)

Abstract. We characterize the nil radicals in some l-rings. Several su�cientconditions for the nil radicals of l-rings to be equal and nilpotent are given. Theresults improve corresponding ones of Birkho� and Pierce. We also show thatthe ideal generated by a nil one-sided ideal which is contained in a nil one-sidedl-ideal is nil.

We mainly describe the relationship and structure of the nil radicals formany lattice-ordered rings (l-rings).

Birkho� and Pierce [1] gave an example of an l-ring R such that the l-radical of R=N(R) is not zero. Therefore the l-radical of such an l-ring isproperly contained in its l-Q radical [2] which coincides with the l-B radical[2], i.e., the l-Q radical and the l-radical of an l-ring are in general, di�erent.However, there are many l-rings for which the l-radical is not only equalto the l-Q radical, but also is equal to the l-nil radical (the sum of all nill-ideals). In this paper we identify some of them and characterize the nilradicals in these l-rings. Several su�cient conditions for the nil radicals ofl-rings to be equal are given.

From Johnson [3] we already know that the l-radical, or the union N(R)of all the nilpotent l-ideals of an l-ring R may not be nilpotent, althoughit must be nil. It also shows that nilpotent and nil in l-rings are di�ernt ingeneral. In the present paper we also discuss the nilpotency of nil l-idealsof l-rings. Birkho� and Pierce [1] pointed out that the l-radical of an l-ringwith either maximum or minimum condition on l-ideals is nilpotent. Wewill prove that if R is an l-ring which satis�es either maximum or minimumcondition on nil l-ideals, then the l-radical, the l-Q radical, and the l-nilradical are identical and nilpotent.

In ring theory the Koethe problem, whether the ideal generated by a nilone-sided ideal is still nil, has not been solved yet. But we show that theideal generated by a nil one-sided ideal contained in a nil one-sided l-ideal isnil, either.

In this paper, R denotes an l-ring. For nonempty subsets X, Y of R,

let XY =�P

n

i=1 xiyi j xi 2 X, yi 2 Y , n is a positive integer, and hXi be

the l-ideal of R generated by X. An f -ring is an l-ring in which a ^ b = 0

� This work was supported by the Hebei Natural Science Foundation.

0236{5294/0/$ 5.00 c 2000 Akad�emiai Kiad�o, Budapest

206 GAO TING

and c = 0 imply ca ^ b = ac ^ b = 0. An important identity satis�ed by anyf -ring is jxyj = jxj jyj. An l-ring satisfying this identity is usually called adistributive l-ring or d-ring. The l-radical N(R) (l-nil radical q(R)) of Ris the sum of all nilpotent (nil) l-ideals of R. An l-ideal A of R is an l-Qideal of R, if for any l-ideal M of R, A� = (A+M)=M 6= f0�g implies thatA� contains a nonzero nilpotent l-ideal of R� = R=M . The sum of all l-Qideals of R, denoted by Q(R), is called the l-Q radical of R [2]. Clearly N(R)

j Q(R) j q(R) j�x 2 R j jxj is nilpotent

j fx 2 R j x is nilpotentg.

Although the Koethe problem in ring theory has not been solved untilnow, in l-rings we have

Theorem 1. (1) If L is a nil left (right) l-ideal of R, then hLRi and

L+ hLRi�hRLi and L+ hRLi

�are nil l-ideals of R, thus the l-ideal of R

generated by any nil one-sided l-ideal of R is nil.(2) The sum of some nil left (right) l-ideals of R is still a nil left (right)

l-ideal of R.

Proof. (1) Without loss of generality we now suppose only that L is anil left l-ideal. For any x 2 hLRi, then there exist a 2 L, r 2 R, such that

jxj 5 ar. Since L is nil, it can be assumed (ra)n = 0, hence from jxjn+1

5 a(ra)nr = 0 we obtain that hLRi is nil.For each x 2 L+ hLRi, then x = x1 + x2, x1 2 L, x2 2 hLRi. Because

L is nil, we may let xm1 = 0, thus xm = xm1 + xm2 +P

� � � x1 � � � x2 � � � = xm2+P

� � �x1 � � �x2 � � � 2 hLRi. In virtue of the fact hLRi is nil, xm is nilpotent,which completes the proof.

(2) It follows immediately from (1).

Corollary. (1) R contains a nonzero nil left l-ideal if and only if Rcontains a nonzero nil right l-ideal.

(2) If A is a nil left (right) ideal of R as a ring and contained in a nilleft (right) l-ideal of R, then the ring ideal A+AR (A+RA) generated byA is a nil ideal of the ring R.

(3) q(R) contains all nil one-sided l-ideals of R, and q�R=q(R)

�= 0.

According to Birkho� and Pierce [1] the l-radical of R=N(R) may havenonzero nilpotent l-ideals, consequently the l-radical and l-Q radical of Rare di�erent in general. But we obtain

Theorem 2. (1) If R is commutative, then N(R) = Q(R) = q(R) =�x 2 R j jxj is nilpotent

.

(2) If R is an f -ring, then N(R) = Q(R) = q(R) = fx 2 R j x is nilpo-tentg.

(3) If R is a d-ring, then Q(R) = q(R) = fx 2 R j x is nilpotentg.

Proof. (1) By commutativity hxi =�y 2 R j jyj 5 njxj+ rjxj, r 2 R,

n is a nonnegative integer. If jxjk = 0, then hxik = 0, which implies that

N(R) j Q(R) j q(R) j�x 2 R j jxj is nilpotent

j N(R). �

Acta Mathematica Hungarica 89, 2000

NIL RADICALS 207

(2) It follows directly from [1, Theorem 16].(3) Let �R = R=Q(R). Clearly �R is a d-ring. Suppose that �a 2 �R+ and �a�R

= �0. Then �a 2 N(�R) j Q(�R) = f�0g by [1, p. 45, De�nition] and [2, Theorem2.1]. Hence �R is an f -ring by [1, pp. 58{59, Lemma 1 and Theorem 14].Suppose that x 2 R and xn = 0. Then �xn = �0 and �x 2 N(�R) = Q(�R) = f�0g,which implies x 2 Q(R). �

Corollary. The ideal generated by a nil one-sided ideal of an f -ringor a d-ring is also nil.

Johnson [3] indicated that the l-radical N(R) of R is nil, but N(R) maynot be nilpotent. This also shows that nil and nilpotent in l-rings are di�er-ent in general.

Next we will consider the problem when a nil l-ideal of R is nilpotent.Birkho� and Pierce [1] gave two su�cient conditions that the l-radical of R isnilpotent. Here we will show some su�cient conditions about nil l-ideals of Rbeing nilpotent. Obviously the results of Birkho� and Pierce are corollariesof our conclusions.

Lemma 1. Suppose that A;B;C are l-ideals of R. Let A(1) = A, A(i)

=A(i�1)A

�, i = 2; 3; : : : ; then

(1)hABiC

�=AhBCi

�, denoted by hABCi.

(2) A(i) =AA(i�1)

�, i = 2; 3; : : : :

(3)A(m)A(n)

�= A(m+n), m;n = 1; 2; : : : :

(4) An j A(n), n = 1; 2; : : : :

We generalize the two results of Birkho� and Pierce to

Theorem 3. If R satis�es the minimum condition on nil l-ideals, thenevery nil l-ideal of R is nilpotent, furthermore N(R) = Q(R) = q(R) is nilpo-tent.

Proof. Suppose that A is a nil l-ideal of R. In the descending chain A k

A(2) k A(3) k � � �, there exists a positive integer m such that A(m) = A(m+1)

by the minimum condition. Let B = A(m), then B(2) =A(m)A(m)

�= A(2m)

= A(m) = B. Assume B 6= 0, let M be a nil l-ideal which is minimal with the

properties: M j B and hBMBi 6= 0. This exists, since hBBBi =B(2)B

�

= hBBi = B(2) = B 6= 0. Then a 2M exists with a = 0 and hBaBi 6= 0. Let

J =�c 2 R j jcj 5 bab0, b; b0 2 B \R+

, then J is an l-ideal of R with 0 6= J

jM , hBJBi 6= 0. Hence J =M by minimality of M . Thus a 5 bab0 forsome b; b0 2 B \R+. From A being nil, there is bn = 0 for a suitable positiveinteger n. It follows that a 5 bab0 5 b2ab02 5 bnab0n = 0, which implies a = 0.This contradicts hBaBi 6= 0, therefore B = 0. Thus A is nilpotent by Am

j A(m) = B. Since q(R) is nil, q(R) is nilpotent, and q(R) j N(R), so N(R)= Q(R) = q(R) is nilpotent.

Acta Mathematica Hungarica 89, 2000

208 GAO TING

Corollary. If R satis�es the minimum condition on l-ideals, then thel-radical of R is nilpotent [1]:

Theorem 4. If R satis�es the maximum condition on nilpotent l-ideals,then

(1) the l-radical N(R) of R is nilpotent.(2) Q(R) = N(R) is nilpotent.

Proof. (1) By the maximum condition, then R contains a maximalnilpotent l-ideal N with Nm = 0 for some positive integer m. Evidently,

N j N(R). Let A be a nilpotent l-ideal of R with Ak = 0, then (A+N)k+m

= 0, and A+N k N . Hence, N = A+N by maximality, which implies thatA j N and N(R) j N , therefore N(R) = N is nilpotent.

(2) Suppose that N(v) is the same as [2, De�nition 2.3]. ApparentlyN(0) = 0 is nilpotent. For each ordinal number v < u, assume that N(v) is anilpotent l-ideal of R. If u is a limit ordinal number, then N(u) =

Pv<u

N(v)is nilpotent by the trans�nite inductive hypothesis and (1); if u = v+1 is nota limit ordinal number, then N(u)=N(v) is the l-radical of R=N(v). SinceR satis�es the maximum condition on nilpotent l-ideals, so does R=N(v).From (1) N(u)=N(v) is nilpotent. By the trans�nite inductive hypothesiswe obtain that N(u) is nilpotent, hence Q(R) is nilpotent via trans�niteinduction and [2, Theorem 3.2], and Q(R) j N(R). Thus Q(R) = N(R).

Corollary. If R satis�es the maximum condition on l-ideals, then thel-radical of R is nilpotent [1]:

Suppose that B is an l-ideal of R, let lB(x) =�r 2 B j jrj jxj = 0

, l(x)

=�r 2 R j jrj jxj = 0

; clearly lB(x) and l(x) are left l-ideals of R.

Lemma 2. Suppose that R is an l-ring with maximum condition on nill-ideals. If R has a nonzero nil l-ideal, then R has a nonzero nilpotent l-ideal.

Proof. Suppose that B is a nonzero nil l-ideal of R, andlB(b)

�

is maximal in�

lB(x)�j 0 < x 2 B

. If bR+ = 0, then bR = 0, and hbi

=�r 2 R j jrj 5 nb+ sb, n is a positive integer, s 2 R

, thus 0 = (Rb)2

= RbR = (b+ bR+Rb+RbR)2, which implies hbi2 = 0. It follows thathbi is nilpotent; if there is r 2 R+ with br > 0, then lB(b) j lB(br), thuslB(b)

�jlB(br)

�. By br 2 bR j B and the maximality of

lB(b)

�we de-

rivelB(b)

�=lB(br)

�. Since B is nil, there exists a positive integer n

such that (br)n = 0 and (br)n�1 6= 0, but (br)n�1 = by, 0 < y 2 R, whence

br 2lB�(br)n�1

��=lB(b)

�, and brb = 0, that is bR+b = 0. Since every

element of R is the di�erence of two positive elements, bRb = 0, so hbi3 = 0,which implies that hbi is nilpotent.

Theorem 5. If R satis�es the maximum condition on nil l-ideals, theneach nil l-ideal of R is nilpotent, and q(R) = Q(R) = N(R) is also nilpotent.

Acta Mathematica Hungarica 89, 2000

NIL RADICALS 209

Proof. Let N be the maximal nilpotent l-ideal of R, A a nil l-idealof R. If A does not contain N , then (A+N)=N is a nonzero nil l-ideal ofR=N , hence from Lemma 2 R=N contains a nonzero nilpotent l-ideal, whichcontradicts the maximality of N . It follows that A j N , namely nil l-idealsof R are nilpotent, and q(R) = Q(R) = N(R) is nilpotent.

Corollary 1. If R satis�es the maximum condition on l-ideals, thenthe l-radical of R is nilpotent [1]:

Corollary 2. (1) If the l-nil radical q(R) of R is commutative, or anf -ring, or satis�es either maximum or minimum condition on nil l-ideals,then N(R) = Q(R) = q(R).

(2) If the l-nil radical q(R) of R is a d-ring, then Q(R) = q(R).(3) If the l-Q radical Q(R) of R satis�es the maximum condition on

nilpotent l-ideals, then N(R) = Q(R).

Proof. Using Theorems 2, 3 and 5, and [2, Theorem 4.4], we have

N(R) = N(R) \ q(R) = N�q(R)

�= q

�q(R)

�= q(R)

= Q�q(R)

�= q(R) \Q(R) = Q(R):

Hence (1) holds. Similarly, we can prove (2) and (3).

Theorem 6. If R is an l-ring with an identity element 1 and the squareof every element of R is positive, then

(1) each nilpotent x of R is, in absolute value, < 1.(2) N(R) = Q(R) = q(R).

Proof. (1) The proof is by induction on the nilpotency index k of x. Fork = 1 the result is trivial. For k = 1 the nilpotency index of x2 is less than k.

Thus x2 = jx2j < 1. Since 0 5 (x� 1)2 = x2 � 2x+ 1 and 0 5 (x+ 1)2 = x2

+2x+1, we have that �(1 + x2) 5 2x 5 (1 + x2). Thus 2jxj = j2xj 5 1+ x2

< 2, and hence jxj < 1.

(2) By (1) A =�x 2 R j jxj 5 n1 for some positive integer n

contains

all of nilpotent elements of R, and hence it contains q(R). For any x; y2 A with jxj 5 n1, jyj 5 m1, we have jx� yj 5 jxj+ jyj 5 (n+m)1, jxyj5 jxj jyj 5 nm1, jx_ yj 5 maxfn;mg1, and jx^ yj 5 minfn;mg1. ThereforeA is the sub-l-ring. Let x^ y = 0, and z 2 A+, then 0 5 z 5 k1 for some pos-itive integer k. Hence by [1] 0 5 zx ^ y 5 kx ^ y = 0. Similarly, xz ^ y = 0.It follows that A is an f -ring, which implies that N(R) = Q(R) = q(R) byCorollary 2(1).

Acta Mathematica Hungarica 89, 2000

210 GAO TING: NIL RADICALS

References

[1] G. Birkho� and R. S. Pierce, Lattice-ordered rings, An. Acad. Brasil Ci., 28 (1956),41{69.

[2] Gao Ting, On the structure of a radical of lattice-ordered rings, J. of Math. Researchand Exp., 17 (1997), 529{537.

[3] D. G. Johnson, A structure theory for a class of lattice-ordered rings, Acta Math., 104(1960), 163{215.

(Received December 28, 1998; revised February 18, 1999)

DEPARTMENT OF MATHEMATICS

HEBEI TEACHERS' UNIVERSITY (EAST CAMPUS)

SHIJIAZHUANG 050016

CHINA

E-MAIL: GAOTING@PUBLIC.SJ.HE.CN

Acta Mathematica Hungarica 89, 2000