Reflexive property of rings. (English) Zbl 1252.16033

Kwak, Tai Keun; Lee, Yang

Commun. Algebra 40, No. 4, 1576-1594 (2012).

As indicated in the title, this paper studies reflexive properties of rings; in particular, how this property and the related property of idempotent reflexiveness transfer between a ring and related rings (trivial extensions, quotient rings, Dorroh extensions, polynomial rings, power series rings, …). Recall, a ring \(R\) is ‘reflexive’ if for all \(a,b\in R\), \(aRb=0\) implies \(bRa=0\). The ring \(R\) is called ‘idempotent reflexive’ if for all \(a,e\in R\) with \(e\) idempotent, \(aRe=0\) if and only if \(eRa=0\).
It is shown that the reflexive condition is Morita invariant. Many examples are provided to illustrate the results but also to show the limitations of others.

Reviewer: Stefan Veldsman (Al-Khodh)

MSC:

16U80	Generalizations of commutativity (associative rings and algebras)
16S36	Ordinary and skew polynomial rings and semigroup rings
16S50	Endomorphism rings; matrix rings
16N60	Prime and semiprime associative rings

Keywords:

reflexive rings; idempotent reflexive rings; matrix rings; polynomial rings; quasi-Baer rings; semiprime rings

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