On annihilator ideals in skew power series rings

Abstract

The purpose of this paper is to study the behavior of the IN and SA property with respect to the skew power series and Laurent series extensions. For an α-compatible ring R it is shown that the skew power series ring $R[[x;\alpha ]]$ is right SA if and only if R is right SA and satisfies the (SQA2) condition. Also it is shown that a semiprime ring R is right SA if and only if $R[[x;\alpha ]]$ is right SA. Moreover, if α is an automorphism, then we have similar results for skew Laurent series ring $R[[x^{\pm 1};\alpha ]].$ Furthermore, if R is a reduced left Noetherian ring, then R is left IN if and only if $R[[x;\alpha ]]$ is left IN if and only if $R[[x^{\pm 1};\alpha ]]$ is left IN. Also we present some examples to show that if we relax each assumption on R, our results can fail.

Keywords:

• Armendariz rings
• Ikeda–Nakayama rings
• SA-rings
• skew Laurent series ring
• skew power series rings

2020 MATHEMATICS SUBJECT CLASSIFICATION::

• Primary: 16D25
• 13F25
• 16S36