Ore extensions of quasi-Baer rings. (English) Zbl 1177.16016

Hong, Chan Yong; Kim, Nam Kyun; Lee, Yang

Commun. Algebra 37, No. 6, 2030-2039 (2009).

Let \(R\) be a ring with 1 and \(S\) a nonempty subset of \(R\). Then \(R\) is called Baer if the right annihilator of \(S\) in \(R\), \(r_R(S)=eR\) for some idempotent \(e\), quasi-Baer if \(r_R(I)=eR\) for every ideal \(I\) of \(R\), and right principally quasi-Baer if \(r_R(I)=eR\) for every principal right ideal \(I\) of \(R\). Let \(\sigma\) be an endomorphism of \(R\), \(\delta\) a \(\sigma\)-derivation of \(R\), and \(R[x;\sigma,\delta]\) the Ore extension of \(R\) (i.e., the polynomial ring over \(x\) such that \(xr=\sigma(r)x+\delta(r)\) for any \(r\in R\)).
Then the authors show some relationships between \(R\) and \(R[x;\sigma,\delta]\). Theorem 1. If \(R\) is quasi-Baer, then so is \(R[x;\sigma,\delta]\).
An idempotent \(e\) is called left semicentral if \(se=ese\) for all \(s\in R\).
Theorem 2. Assume \(\sigma(e)=e\) for any left semicentral idempotent \(e\in R\). If \(R\) is right principally quasi-Baer, then so is \(R[x;\sigma,\delta]\).
Theorem 3. Let \(R\) be a semiprime ring with \(\sigma(I)\subset I\) for any ideal \(I\) of \(R\). If \(R[x;\sigma,\delta]\) is quasi-Baer (right principally quasi-Baer), then so is \(R\).

Reviewer: George Szeto (Peoria)

MSC:

16S36	Ordinary and skew polynomial rings and semigroup rings
16P60	Chain conditions on annihilators and summands: Goldie-type conditions
16N60	Prime and semiprime associative rings
16D25	Ideals in associative algebras

Keywords:

Ore extensions; quasi-Baer rings; semiprime rings; semicentral idempotents