On extended reversible rings. (English) Zbl 1167.16018

Başer, Muhittin; Hong, Chan Yong; Kwak, Tai Keun

Algebra Colloq. 16, No. 1, 37-48 (2009).

Let \(R\) be a ring with 1. An endomorphism \(\alpha\) of \(R\) is called right (resp. left) reversible if whenever \(ab=0\) for \(a,b\in R\), \(b\alpha(a)=0\) (resp. \(\alpha(b)a=0\)). A ring \(R\) is called right (resp. left) \(\alpha\)-reversible if there exists a right (resp. left) reversible endomorphism \(\alpha\) of \(R\), and \(R\) is \(\alpha\)-reversible if it is both right and left \(\alpha\)-reversible ring. Then the authors show some properties of a right \(\alpha\)-, and left \(\alpha\)-, and \(\alpha\)-reversible ring. Let \(R[x;\alpha]\) be the skew polynomial ring. A ring \(R\) is called \(\alpha\)-Armendariz (resp. \(\alpha\)-skew Armendariz) if for \(p=\sum_{i=0}^m a_ix^i\) and \(q=\sum_{j=0}^n b_jx^j\in R[x;\alpha]\), \(pq=0\) implies that \(a_ib_j=0\) (resp. \(a_i\alpha^i(b_j)=0\)) for all \(0\leq i\leq m\) and \(0\leq j\leq n\). Then some relations are given between the class of \(\alpha\)-reversible rings and other classes of rings such as \(\alpha\)-skew Armendariz rings, skew polynomial rings, reduced rings, and rings of Laurent polynomials \(R[x;x^{-1}]\).
Theorem 1. (1) If \(R\) is a reduced and \(\alpha\)-reversible ring, then \(R\) is \(\alpha\)-skew Armendariz. (2) If \(R[x;\alpha]\) is a reversible ring, then \(R\) is \(\alpha\)-reversible.
Theorem 2. (1) The polynomial ring \(R[x]\) is \(\alpha\)-reversible ring if and only if \(R[x;x^{-1}]\) is right \(\alpha\)-reversible. (2) Let \(R\) be an Armendariz ring. Then \(R\) is right \(\alpha\)-reversible if and only if \(R[x]\) is right \(\alpha\)-reversible. (3) Let \(R\) be a reduced ring and \(n\) a positive integer. If \(R\) is right \(\alpha\)-reversible with \(\alpha(1)=1\), then \(R[x]/\langle x^n\rangle\) is right \(\overline\alpha\)-reversible ring, where \(\langle x^n\rangle\) is the ideal generated by \(x^n\) and \(\overline\alpha\) is the endomorphism of \(R[x]/\langle x^n\rangle\) such that \(\overline\alpha(f+\langle x^n\rangle)=\alpha(f)+\langle x^n\rangle\). Thus some known results are derived.

Reviewer: George Szeto (Peoria)

MSC:

16S36	Ordinary and skew polynomial rings and semigroup rings
16W20	Automorphisms and endomorphisms
16U80	Generalizations of commutativity (associative rings and algebras)
16W60	Valuations, completions, formal power series and related constructions (associative rings and algebras)

Keywords:

reduced rings; reversible rings; generalized Armendariz rings; skew polynomial rings; skew Armendariz rings; rings of Laurent polynomials; endomorphisms

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