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On a property of polynomial rings over reversible rings. (English) Zbl 1448.16037
Summary: In this article, we first observe a sort of property of ideals generated by zero-dividing polynomials over reversible rings, in relation with products of ideals at zero. As a natural consequence of this result, we introduce the concept of a partially reflexive ring that is related to the reflexive ring property. It is shown that abelian \( \pi \)-regular rings are partially reflexive when Jacobson radicals are nilpotent. A ring \( R \), in which the Jacobson radical \( J(R) \) is nilpotent and \( R/J(R) \) is simple, is shown to be partially reflexive; and it is proved that the polynomial ring over \( R \) is also partially reflexive. The structures of several kinds of algebraic systems are investigated with respect to the partial reflexivity.
MSC:
16S36 Ordinary and skew polynomial rings and semigroup rings
16N20 Jacobson radical, quasimultiplication
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