On skew polynomials over Ikeda-Nakayama rings

Abstract

A ring R is called a left Ikeda-Nakayama ring (left IN-ring) if the right annihilator of the intersection of any two left ideals is the sum of the two right annihilators. Also a ring R is called a right SA-ring if the sum of right annihilators of two ideals is a right annihilator of an ideal of R. In this paper for a compatible endomorphism α of R, we show that: (i) If $R[x;\alpha ]$ is a left IN-ring, then R is an Armendariz left IN-ring. (ii) If R is a reduced left IN-ring with finitely many minimal prime ideals, then $R[x;\alpha ]$ is a left IN-ring. (iii) $R[x;\alpha ]$ is a right SA-ring, if and only if R is a quasi-Armendariz right SA-ring. We give a class of non-reduced rings R such that $R[x]$ is left IN-ring. Also we give some examples to show that assumption compatibility on α is not superfluous.

Keywords:

• Armendariz rings
• Ikeda-Nakayama rings
• quasi-Armendariz rings
• SA-rings
• skew polynomials rings

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 16D25
• 16S36

Acknowledgments

The authors would like to thank the editor and the anonymous referee for many constructive comments that helped improve the quality of the paper.