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

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 $$R$$.

### 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
Full Text: