Moroccan Journal of Algebra and Geometry with Applications

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Formal power series rings with one absorbing factorization

Abstract

Let $R$ be a commutative ring with identity. A proper ideal $I$ of $R$ is said to be 1-absorbing prime ideal if whenever $xyz \in I$ for some nonunit elements $x,y,z\in R,$ then either $xy\in I$ or $z\in I.$ The ring $R$ is called a 1-absorbing prime factorization ring (OAF-ring) if every proper ideal has an OA-factorization. In this note, we characterize commutative rings $R$ (respectively, ring extensions $A\subset B$) for which the ring of formal power series $R[[X]]$ (respectively, the ring $A+XB[X]$ or $A+XB[[X]]$) is an OAF-ring.

Keywords:  OA-ideals, OAF-rings, formal power series rings.

MSC numbers: 13A15, 13B25, 3F25, 13F15, 13B99.

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