zbMATH — the first resource for mathematics

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.

MSC:
 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
Full Text: