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McCoy rings relative to a monoid. (English) Zbl 1207.16041
Let \(R\) be a ring with 1 and \(M\) a monoid. Then \(R\) is called a right \(M\)-McCoy ring if whenever \(\alpha=a_1g_1+\cdots+a_ng_n\neq 0\) and \(\beta=b_1h_1+\cdots+b_mh_m\neq 0\in R[M]\) where \(g_i,h_j\in M\) and \(a_i,b_j\in R\) such that \(\alpha\beta=0\), then \(\alpha r=0\) for some nonzero \(r\in R\). In particular, \(R\) is called right linearly \(M\)-McCoy when \(n=m=2\). A left \(M\)-McCoy ring is similarly defined, and an \(M\)-McCoy ring is both left and right \(M\)-McCoy.
A monoid \(M\) is called a u.p.-monoid if for any finite subsets \(A,B\subset M\), there exists an element \(g\in M\) such that \(g=ab\) for some unique elements \(a\in A\) and \(b\in B\).
Then an \(M\)-McCoy ring is a generalization of a McCoy ring and the author gives \(M\)-McCoy rings \(R\) for different classes of rings \(R\) and monoids \(M\): (1) Let \(R\) be a reversible ring (i.e., \(ab=0\) implies \(ba=0\) for all \(a,b\in R\)), and \(M\) a u.p.-monoid. Then \(R\) is \(M\)-McCoy; (2) Let \(R\) be a right duo ring (i.e., all right and left ideals of \(R\) are ideals of \(R\)), and \(M\) a strictly totally ordered monoid. Then \(R\) is right \(M\)-McCoy; (3) Let \(R\) be a semicommutative ring (i.e., \(ab=0\) implies \(aRb=0\) for all \(a,b\in R\)), and \(M\) a u.p.-monoid. Then \(R\) is right linearly \(M\)-McCoy.
Moreover, properties of \(R[M]\) are also shown for a semicommutative ring \(R\), a 2-primal ring \(R\), and a u.p.-monoid \(M\).

MSC:
16U80 Generalizations of commutativity (associative rings and algebras)
16S36 Ordinary and skew polynomial rings and semigroup rings
20M25 Semigroup rings, multiplicative semigroups of rings
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
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