Rings over which every semi-primary ideal is 1-absorbing primary

Abstract

Let R be commutative ring with $1\ne 0$. A proper ideal I of R is called a 1-absorbing primary ideal of R if whenever nonunit elements $a,b,c\in R$ and $\mathit{abc}\in I$, then $ab\in I$ or $c\in \sqrt{I}$. It is proved that every primary ideal of R is 1-absorbing primary and every 1-absorbing primary ideal of R is semi-primary (that is ideals with prime radical). However, these three concepts are different. In this paper, we characterize rings R over which every semi-primary ideal is 1-absorbing primary and (resp. Noetherian) rings R over which every 1-absorbing primary ideal is prime (resp. primary). Many examples are given to illustrate the obtained results.

Keywords:

• 1-Absorbing primary ideals
• primary ideals
• semi-primary ideals

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A15
• 13A99

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Acknowledgments

The authors would like to thank the referee for careful reading of the manuscript.

Additional information

Funding

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under Grant number R.G.P.1/178/41.