Annihilator Ideals of Noncommutative Ring Constructions

作者：Alhevaz Abdollah; Kiani Dariush^*

来源：Communications in Algebra, 2016, 44(1): 63-86.

DOI：10.1080/00927872.2014.937536

• Quasi-Armendariz ring
• Quasi-Baer ring
• Quasi-Gaussian ring
• Reflexive ideal
• Skew monoid ring
• Skew power series ring
• Weakly-rigid ideal

Hirano studied the quasi-Armendariz property of rings, and then this concept was generalized by some authors, defining quasi-Armendariz property for skew polynomial rings and monoid rings. In this article, we consider unified approach to the quasi-Armendariz property of skew power series rings and skew polynomial rings by considering the quasi-Armendariz condition in mixed extension ring [R; I][x; sigma], introducing the concept of so-called (sigma, I)-quasi Armendariz ring, where R is an associative ring equipped with an endomorphism sigma and I is an sigma-stable ideal of R. We study the ring-theoretical properties of (sigma, I)-quasi Armendariz rings, and we obtain various necessary or sufficient conditions for a ring to be (sigma, I)-quasi Armendariz. Constructing various examples, we classify how the (sigma, I)-quasi Armendariz property behaves under various ring extensions. Furthermore, we show that a number of interesting properties of an (sigma, I)-quasi Armendariz ring R such as reflexive and quasi-Baer property transfer to its mixed extension ring and vice versa. In this way, we extend the well-known results about quasi-Armendariz property in ordinary polynomial rings and skew polynomial rings for this class of mixed extensions. We pay also a particular attention to quasi-Gaussian rings.

• 出版日期2016