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A characterization of primal Noetherian rings. (English) Zbl 0703.16001

An ideal A of a ring R (with identity) is called primal if there exists \(c\in R\setminus A\) with c(\(\sum B_ i)\subseteq A\) provided \(c_ iB_ i\subseteq A\) for some \(c_ i\in R\setminus A\) where the \(B_ i's\) are two-sided ideals of R. A ring R is called primal if all ideals of R are primal. It is proved (in the first part of Theorem 1) that the prime ideals of R are in a chain whenever R is primal, however the proof of the converse for noetherian rings appears to be not complete.
(The following is a proof of the second part of Theorem 1 in the paper under review. Let R be a ring with identity and maximum condition for two-sided ideals such that the set of prime ideals of R is totally ordered by set inclusion. Then there exists for each proper ideal A of R a unique prime ideal P of R with \(P^ n\subseteq A\subseteq P\) for some n. To show that A is primal assume \(c_ iB_ i\subseteq A\), \(c_ i\not\in A\). Then \(P_ i^{n_ i}\subseteq B_ i\subseteq P_ i\) and from \(n_ 0\) minimal with \(c_ iP_ i^{n_ 0}=(c_ iP_ i^{n_ 0-1})P_ i\subseteq A\) it follows that there exists \(d_ i\in c_ iP_ i^{n_ 0-1}\), \(d_ i\not\in A\) with \(d_ iP_ i\subseteq A\). Hence, \(\sum B_ i\subseteq P\) with P maximal among the prime ideals \(P_ i\), and A is primal, R is primal.)
Reviewer: H.H.Brungs

MSC:

16D25 Ideals in associative algebras
16P40 Noetherian rings and modules (associative rings and algebras)
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