|
|
| |
| Artinian Radical and its Application |
| |
Citation： |
Xu Yonghua.Artinian Radical and its Application[J].Chinese Annals of Mathematics B,1985,6(2):215~228 |
| Page view： 725
Net amount： 716 |
Authors： |
Xu Yonghua; |
|
|
| Abstract： |
Let R be a left and right Neotherian ring with identity. Let A be the Artinian radical. Lenagan^[3] pointed out that R has Artinian quotient ring if A=0 and the Krull dimension of R is one. In this paper first the structure of Artinian radical is investigated. Then for R with Krull dimension one the author gives a necessary and sufficient condition under which R has Artinian quotient ring. The main results are as follows: (i) A=eR, where e is a central idempotent element of R, if and only if r(A)^\lambda=l(A)^\lambda=([\mathop \cap \limits_{\scriptstylei = 1, \cdots ,{n_k}\hfill\atop
\scriptstylek = 1, \cdots ,m\hfill} p(a_i^{(k)})\])^\lambda where \lambda is a positive integer, p(a_i^(k)) are prime ideals of R and r(A)(lA)) is the notation of right(:eft) annihilatorof A (see Theorem 7). (ii) In the case (i) R=A\oplus(A)^\lambda. (iii) If R has Krull dimension one, then R has Artinian quotient ring if and only if there exists a positive integer \lambda such that r(A)^\lambda=l(A)^\lambda= ([\mathop \cap \limits_{\scriptstylei = 1, \cdots ,{n_k}\hfill\atop
\scriptstylek = 0, \cdots ,m\hfill} p(a_i^{(k)})\] )^\lambda. |
Keywords： |
|
Classification： |
|
|
|
Download PDF Full-Text
|
|
|
|
