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

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)].

### 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

Zbl 0077.25903
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### References:

 [1] Passmann, The algebraic structure of group rings (1977) [2] Okninski, Semigroup algebras (1991) [3] DOI: 10.1016/j.jalgebra.2005.10.008 · Zbl 1110.16036 · doi:10.1016/j.jalgebra.2005.10.008 [4] DOI: 10.2307/2309082 · Zbl 0077.25903 · doi:10.2307/2309082 [5] DOI: 10.2307/2307526 · doi:10.2307/2307526 [6] DOI: 10.1016/S0022-4049(01)00053-6 · Zbl 1007.16020 · doi:10.1016/S0022-4049(01)00053-6 [7] Gilmer, J. Reine Angew. Math. 278 pp 145– (1975) [8] DOI: 10.2307/2036469 · Zbl 0219.13023 · doi:10.2307/2036469 [9] DOI: 10.2307/2303094 · Zbl 0060.07703 · doi:10.2307/2303094
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