Frontiers of Mathematics in China >

2016 , Vol. 11 >Issue 4: 869 - 900

DOI: https://doi.org/10.1007/s11464-016-0552-9

RESEARCH ARTICLE

On extensions of matrix rings with skew Hochschild 2-cocycles

  • Chan Yong HONG 1 ,
  • Nam Kyun KIM , 2 ,
  • Tai Keun KWAK 3 ,
  • Yang LEE 4
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  • 1. Department of Mathematics and Research Institute for Basic Sciences, Department of Mathematics and Research Institute for Basic Sciences,
  • 2. School of Basic Sciences, Hanbat National University, Daejeon 34158, Korea
  • 3. Department of Mathematics, Daejin University, Pocheon 11159, Korea
  • 4. Department of Mathematics, Pusan National University, Pusan 46241, Korea

Received date: 16 Sep 2015

Accepted date: 26 Apr 2016

Published date: 30 Aug 2016

Copyright

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg
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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.

Key words： Skew Hochschild extensions; matrix rings; skew triangular matrix rings; (uniquely) clean rings; symmetric rings

Cite this article

Chan Yong HONG , Nam Kyun KIM , Tai Keun KWAK , Yang LEE . On extensions of matrix rings with skew Hochschild 2-cocycles[J]. Frontiers of Mathematics in China, 2016 , 11(4) : 869 -900 . DOI: 10.1007/s11464-016-0552-9

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