We first consider properties and basic extensions of symmetric rings. We next argue about the symmetry of some kinds of polynomial rings, and show that if R is a reduced ring then R[x]/（x^n） is a symmetric ring, whe...We first consider properties and basic extensions of symmetric rings. We next argue about the symmetry of some kinds of polynomial rings, and show that if R is a reduced ring then R[x]/（x^n） is a symmetric ring, where （x^n） is the ideal generated by x^n and n is a positive integer. Consequently, we prove that for a right Ore ring R with Q its classical right quotient ring, R is symmetric if and only if Q is symmetric.展开更多
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 exam...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.展开更多
文摘We first consider properties and basic extensions of symmetric rings. We next argue about the symmetry of some kinds of polynomial rings, and show that if R is a reduced ring then R[x]/（x^n） is a symmetric ring, where （x^n） is the ideal generated by x^n and n is a positive integer. Consequently, we prove that for a right Ore ring R with Q its classical right quotient ring, R is symmetric if and only if Q is symmetric.
文摘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.
