摘要
We study structures of endomorphisms and introduce a skew Hochschild 2-cocycles related to 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.
We study structures of endomorphisms and introduce a skew Hochschild 2-cocycles related to 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.
