On extensions of matrix rings with skew Hochschild 2-cocycles

Abstract

We study structures of Hochschild 2-cocycles related to endomorphisms and introduce a skew Hochschild 2-cocycle. We moreover define skew Hochschild extensions equipped with skew Hochschild 2-cocycles, and then we examine uniquely clean, Abelian, directly finite, symmetric, and reversible ring properties of skew Hochschild extensions and related ring systems. The results obtained here provide various kinds of examples of such rings. Especially, we give an answer negatively to the question of H. Lin and C. Xi for the corresponding Hochschild extensions of reversible (or semicommutative) rings. Finally, we establish three kinds of Hochschild extensions with Hochschild 2-cocycles and skew Hochschild 2-cocycles.Keywords Skew Hochschild extensions, matrix rings, skew triangular matrix rings, (uniquely) clean rings, symmetric rings MSC 16S50, 15A30, 15B33 Throughout this paper, R denotes an associative ring with identity. We use R[x] to denote the polynomial ring with an indeterminate x over R. For the factor ring R[x]/(x n+1 ), (x n+1 ) is the ideal of R[x] generated by x n+1 for n 1. Z (Z n ) denotes the ring of integers (modulo n).Following the literature, a ring is called reduced if it has no nonzero nilpotent elements. A ring R is called symmetric if abc = 0 implies acb = 0 for a, b, c ∈ R. A ring R is called reversible if ab = 0 implies ba = 0 for a, b ∈ R. A ring R is called semicommutative if ab = 0 implies aRb = 0 for a, b ∈ R. Clearly, reduced rings are symmetric, symmetric rings are reversible, and reversible rings are semicommutative, but the converse is not true in either case. A ring R is called