McCoy condition on ideals of coefficients. (Q2872179)

Let \(R\) be a ring with \(1\), \(N(R)\) the set of nilpotent elements in \(R\), and \(R[x]\) the polynomial ring of an indeterminate \(x\) over \(R\). Then \(R\) is called a right McCoy ring if \(f(x)g(x)=0\) for nonzero \(f(x),g(x)\in R[x]\) implies \(f(x)r=0\) for some nonzero \(r\in R\). A ring \(R\) is called an ideal-\(\pi\)-McCoy ring if \(f(x)g(x)\in N(R[x])\) implies \(f(x)a\in N(R[x])\) for some nonzero \(a\in J\) where \(J\) is the ideal of \(R\) generated by the coefficients of \(g(x)\).NEWLINENEWLINE The authors show some basic ring theoretic properties of an ideal-\(\pi\)-McCoy ring \(R\) and relations between \(R\) and other classes of rings such as \(R[x]\), \(R[x]/(x^n)\) where \((x^n)\) is the ideal of \(R[x]\) generated by \(x^n\) for any \(n\geq 2\), matrix rings over \(R\), Armendariz rings, and the classical right quotient ring \(Q\) of a right Ore ring \(R\). (1) Let \(e\) be a central idempotent of a ring \(R\). Then \(R\) is ideal-\(\pi\)-McCoy if and only if so are \(Re\) and \(R(1-e)\). (2) The class of ideal-\(\pi\)-McCoy rings is closed under direct limits with injective maps. (3) If \(R[x]\) is ideal-\(\pi\)-McCoy over a ring \(R\), then so is \(R\). (4) Let \(R\) be a right Ore ring with its classical right quotient ring \(Q\). If \(R\) is ideal-\(\pi\)-McCoy such that non-regular polynomials in \(R[x]\) are nilpotent, then \(Q\) is ideal-\(\pi\)-McCoy where an element is called regular if its right and left annihilators are \(0\).