Extensions of McCoy’s theorem. (English) Zbl 1195.16026

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

Glasg. Math. J. 52, No. 1, 155-159 (2010).

Let \(R\) be a ring with 1, \(S\) a ring extension of \(R\), and \(r_R(A)\) the right annihilator of a subset \(A\) of \(S\) in \(R\). Let \(\sigma\) be an automorphism and \(\delta\) a \(\sigma\)-derivation of \(R\), and \(S\) (\(=R[x;\sigma,\delta]\)) the Ore extension of \(R\).
The authors show that for a right ideal \(A\) of \(S\), \(r_S(A)\neq 0\) implies \(r_R(A)\neq 0\).
Moreover, a monoid \(G\) is called a unique product monoid if for any two non-empty finite subsets \(A,B\subset G\), there exists a \(c\in G\) uniquely presented in the form \(ab\) where \(a\in A\) and \(b\in B\). Assume \(G\) acts on \(R\) by means of a homomorphism \(\sigma\) into the automorphism group of \(R\). Let \(R*G\) be the skew monoid ring; that is, it is a left \(R\)-module with a free basis \(\{g\mid g\in G\}\) and \(gr=\sigma_g(r)g\) for \(r\in R\) and \(\sigma_g(r)\in R\). Then \(r_{R*G}(A)\neq 0\) implies \(r_R(A)\neq 0\) for a right ideal \(A\) of \(R*G\).
The same result also holds for power series rings \(R[\![x;\sigma]\!]\) or \(R[\![x,x^{-1};\sigma]\!]\) over a semiprime ring \(R\). This result was shown by N. H. McCoy for \(R[x_1,x_2,\dots,x_k]\) [in Am. Math. Mon. 64, 28-29 (1957; Zbl 0077.25903)].

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
16W60	Valuations, completions, formal power series and related constructions (associative rings and algebras)
16W20	Automorphisms and endomorphisms

Keywords:

Ore extensions; quasi-Baer rings; semiprime rings; right annihilators; right ideals; unique product monoids; skew monoid rings; power series rings

Citations:

Zbl 0077.25903

References:

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