Internat. J. Math. & Math. Scl.
Vol. 8 No. (1985) 205-207
2O5
RINGS
DECOMPOSED INTO DIRECT SUMS
OF J-RINGS ANDNIL RINGS
HISAO TOMINAGA Department of Mathematics
Okayama University Okayama 700, Japan (Received April 25, 1984)
ABSTRACT. Let R be a ring (not necessarily with identity) and let E denote the set of idempotents of R. We prove that R is a direct sum of a J-ring (every element is a power of itself) and a nil ring if and only if R is strongly H-regular and E is con- tained in some J-ideal of R. As a direct consequence of this result, the main theorem of [i] follows.
KEY WORDS AND PHRASES. Periodic, potent, J-ring, niZ ring, strongly H-regular ring, direct sum.
/980 MATHEMATICS SUBJECT CLASSIFICATION CODE. Primary 16A15; Secondary 16A70.
i. INTRODUCTION.
’hroughout the present note, R will represent a ring (not necessarily with iden-
tity), N the set of nilpotent elements of R, and E the set of idempotents of R. We say that R is periodic if for each r R, there exist distinct positive integers h, k
h k
for which r =r According to Chacron’s theorem (see, e.g., [2, Theorem i]), R is periodic if and only if for each r e R, there exists a polynomial f(A) with integer coefficients such that
r-r2f(r)
N. An element r of R is called potent if there is an integer n i such that rn r. We denote by I the set of potent elements of R. If R coincides with I, R is called a J-ring. As is well known, every J-ring is commuta- tive (Jacobson’s theorem). An ideal of R is called a J-ideal if it is a J-ring. Also, we denote by I0 the set {r R r generates a subring with identity}. It is clear that E ! I ! IO. Furthermore, if I
0 is a subring of R then I
0 coincides with I. In fact, if r is an arbitrary element of I
0 then there exists a polynomial f() with r2
integer coefficients such that r f(r). This proves that I
0 is a reduced periodic ring, and therefore a J-ring. Especially, R is a J-ring if and only if R =I
O. If R
is the direct sum of a J-ideal I’ and a nil ideal N’, then it is easy to see that I’
I =I and N’ =N.
0
2. MAIN THEOREM.
Now, the main theorem of this note is stated as follows"
206 H. TOMINAGA
’?liEOREM I. The following conditions are equivalent"
i] R is right (or left) n-regular and E is contained in some J-ideal A of R.
2 R is periodic and E is contained in some reduced ideal A of R.
3) R is a direct sum of a J-ring and a nil ring.
Mc)r’e [)recisely, if i) or 2) is satisfied, then N is an ideal of R, R=A e N, and A=
I =I0. In particular, if R is right (or left) s-unital, that is, r rR (or r Rr)
for all r R, then each of i), 2) is equivalent to 4) R is a J-ring.
PROOF. Obviously, 3) 2) i).
i) 3). By a result of Dischinger (see, e.g., [3, Proposition 2]), R is strongly w-regular. Let r be an arbitrary element of R. Then there exists a positive integer
r2n
s,,r2n
n 2n and elements s’, s" of R such that =s =r We put s
=rns
As is easilyseen,
s
s"rns’ s"2r
nand
n ,rn s" 2n n 2n ,,rn
r s r r r s
rns
Hence,
n n
,,ms s,,r2ns
,,rn,,ms
n nr s=r s
=rns
=s =s r =stand
2n n n n n n
r s =r sr =r s r =r
n n n
Since e =r s is an idempotent with re =er ( A) and r e =r we see that
(r-re)n
=rn(l_e)n=0.This together with r=re +(r-re) proves that r is represented as a sum of an element in A and a nilpotent element. Now, let a, b e A, and x, y e N. Noting that xa.yb xyba and ax.by=baxy, we can easily see that xa e N n A=0 and ax=0; NA=AN=O. Set xy=c+u and x+y=d+v (c, d e A and u, v e N), where we may assume that u =v =0.
In view of NA=0, we obtain
(x)2
xy(c +u) =xuand
(x+
y)2
(x +y)(d+v)=(x+y)v,and therefore
(xy)+I xyu 0 and
(x+y)+l= (x+y)v=O.
,’e have thus seen that N forms an ideal of R and R =A N.
RINGS DECOMPOSED INTO DIRECT SUMS OF J-RINGS AND NIL RINGS 207
Given an integer n 1 we denote by In the set {r R
rn
=r} In [i] Abu-Khuzam and Yaqub proved that if R is a periodic ring with N commutative and for which I forms an ideal, then R is a subdirect sum of finite fields of at most n elements
n
and a nil commutative ring. The next corollary includes this result.
COROLLARY i. If R is periodic and I forms an ideal of R for some integer n n
then R =I N and I is a subdirect sum of finite fields of at most n elements.
REFERENCES
i. ABU-KHUZAM, H. and YAQUB, A. Periodic rings with commuting nilpotents, Internat.
J. Math. Math. Sci. 7 (1984) 403-406.
2. BELL, }{.E. On commutativity of periodic rings and near rings, Acta Math. Acad.
Sci. Hungar. 36 (1980), 293-302.
3. HIRANO, Y. Some studies on strongly w-regular rings, Math. J. Okayama Univ. 2--0 (1978), 141-149.
Journal of Applied Mathematics and Decision Sciences
Special Issue on
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Call for Papers
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However, more and more researchers have found that the financial environment is not ruled by mathematical distribu- tions or statistical models. In such situations, some attempts have also been made to develop financial engineering mod- els using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estima- tion technique which does not make any distributional as- sumptions regarding the underlying asset. Instead, ANN ap- proach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting pa- rameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior perfor- mance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should es- pecially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelli- gent, easy-to-use, and/or comprehensible computational sys- tems (e.g., decision support systems, agent-based system, and web-based systems)
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