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Rings with involution and chain conditions on bi-ideals. (English. Russian original) Zbl 0826.16031
Russ. Math. Surv. 48, No. 5, 161-162 (1993); translation from Usp. Mat. Nauk 48, No. 5(293), 159-160 (1993).
Let \(A\) be a not necessarily associative ring with an involution \(*\), \(S\) be a subset of the ring \(A\). Let \([S\rangle\) (respectively \(\langle S]\)) denote the right (respectively left) ideal of \(A\) generated by \(S\). The subset \(S\) is called a bi-ideal of \(A\) if it is a subring and \(S \supseteq [S \rangle \langle S]\). Further, a bi-ideal \(S\) is called a \(*\)-bi-ideal if \(S^* \subseteq S\).
Rings with maximum and minimum conditions on \(*\)-bi-ideals are investigated. Associative rings with involution satisfying the minimum condition on \(*\)-bi-ideals are characterized (Theorem 1, Corollary 2). In contrast with the Hilbert basis theorem, for a polynomial ring \(A[x]\) the maximum condition on \(*\)-bi-ideals is equivalent to a stronger condition on the (not necessarily associative) coefficient ring \(A\) than the maximum condition on \(*\)-bi-ideals (Theorem 3, Corollary 4). The paper does not contain the proofs of these results.

16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16P70 Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras)
17A30 Nonassociative algebras satisfying other identities
16D25 Ideals in associative algebras
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