Two absorbing factorization formal power series rings

Abstract

Let R be a commutative ring with identity. A proper ideal I of R is said to be 2-absorbing if whenever $\mathit{\text{xyz}}\in I$ for some elements $x,y,z\in R,$ then either $\mathit{\text{xy}}\in I$ or $\mathit{\text{xz}}\in I$ or $\mathit{\text{yz}}\in I.$ The ring R is called a two-absorbing factorization ring (TAF-ring) if every proper ideal of R has a two absorbing-factorization that is every ideal is a product of 2-absorbing ideals. In this note, we characterize commutative rings R (respectively, commutative ring extensions $A\subset B$) for which the ring of formal power series $R[[X]]$ (respectively, the ring $A+\mathit{\text{XB}}[[X]]$) is a TAF-ring.

KEYWORDS:

• Formal power series rings
• TAF-rings
• two absorbing ideals

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A15
• 13F25
• 13F15
• 13B99

Acknowledgments

The author would like to thank the referee for his/her useful comments.