On skew Armendariz rings. (English) Zbl 1042.16014

Hong, Chan Yong; Kim, Nam Kyun; Kwak, Tai Keun

Commun. Algebra 31, No. 1, 103-122 (2003).

Throughout the paper \(R\) denotes an associative ring with identity. For a ring endomorphism \(\alpha\colon R\to R\), the skew polynomial ring \(R[x;\alpha]\) of \(R\) is the ring obtained by giving the polynomial ring over \(R\) the new multiplication \(xr=\alpha(r)x\) for all \(r\in R\). A Baer ring is a ring in which the right (left) annihilator of every non-empty subset is generated by an idempotent. The authors introduce the following notion: for an endomorphism \(\alpha\colon R\to R\), \(R\) is called a skew Armendariz ring with the endomorphism \(\alpha\) (simply, an \(\alpha\)-skew Armendariz ring) if for \(p=\sum_{i=0}^ma_ix^i\), and \(q=\sum_{j=0}^nb_jx^j\) in \(R[x;\alpha]\), \(pq=0\) implies \(a_i\alpha^i(b_j)=0\) for all \(0\leq i\leq m\), and \(0\leq j\leq n\).
If \(\alpha\) is an endomorphism of \(R\) such that \(\alpha^t=I_R\) for some positive integer \(t\) and the identity endomorphism \(I_R\) of \(R\), then it is shown that \(R\) is \(\alpha\)-skew Armendariz if and only if \(R[x]\) is \(\alpha\)-skew Armendariz. Also, any domain \(R\) is \(\alpha\)-skew Armendariz for any endomorphism \(\alpha\) of \(R\). The authors prove that, for an \(\alpha\)-skew Armendariz ring \(R\), \(R[x;\alpha]\) is Abelian if and only if \(\alpha(e)=e\) for any \(e^2=e\in R\). Finally, if \(\alpha\) is an automorphism of \(R\) with \(\alpha(e)=e\) for any \(e^2=e\in R\) and \(R\) is an \(\alpha\)-skew Armendariz ring, then \(R\) is a Baer ring if and only if \(R[x;\alpha]\) is a Baer ring.

Reviewer: Iuliu Crivei (Cluj-Napoca)

MSC:

16S36	Ordinary and skew polynomial rings and semigroup rings
16W20	Automorphisms and endomorphisms

Keywords:

Armendariz rings; skew polynomial rings; Baer rings; annihilators; idempotents; endomorphisms

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