Extensions of reversible rings. (English) Zbl 1040.16021

Kim, Nam Kyun; Lee, Yang

J. Pure Appl. Algebra 185, No. 1-3, 207-223 (2003).

A ring \(R\) is called reversible if \(ab = 0\) implies \(ba = 0\) for \(a,b\) \(\in R\). Some authors call this ring zero-commutative. The authors obtain some basic properties of basic extensions of these rings. Let \(T(R,R)\) and \(R[x]\) be the \(2\) by \(2\) upper triangular matrix ring and polynomial ring over \(R\), respectively. If \(R\) is reduced (i.e., there are no non-zero nilpotent elements in \(R\)), the authors prove that \(T(R,R)\) and \(R[x]/ (x^n)\) are reversible rings, where \((x^n)\) is the ideal of \(R[x]\) generated by \(x^n\).

Reviewer: Xue Weimin (Fujian)

MSC:

16U80	Generalizations of commutativity (associative rings and algebras)
16U10	Integral domains (associative rings and algebras)
16U20	Ore rings, multiplicative sets, Ore localization

Keywords:

reversible rings; zero-commutative rings; reduced rings

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