Extensions of linearly McCoy rings. (Q2872989)

Let \(R\) be a ring with \(1\) and \(R[x]\) the polynomial ring with an indeterminate \(x\) over \(R\). Then \(R\) is called right linearly McCoy whenever linear polynomials \(f(x),g(x)\in R[x]\) are such that \(f(x)g(x)=0\) implies that \(f(x)r=0=sg(x)\) for some non-zero \(r,s\in R\). Similarly, a left and a two sided linearly McCoy ring are defined.NEWLINENEWLINE The authors show some extension properties of linearly McCoy rings. (1) There exists a semi-commutative ring \(R\) such that \(R[x]\) is not linearly McCoy where \(R\) is called semi-commutative if \(ab=0\) implies \(aRb=0\) for \(a,b\in R\). (2) Let \(R_n\) be the ring of the \(n\times n\)-upper triangular matrices over a ring \(R\) with the same element along the main diagonal for \(n\geq 1\). Then \(R\) is linearly McCoy if and only if so is \(R_n\). (3) If \(R\) is linearly McCoy, then so is \(R[x]/(x^n)\) where \((x^n)\) is the ideal generated by \(x^n\) for any positive integer \(n\). (4) Suppose \(R\) has the classical right quotient ring \(Q\). Then \(R\) is right linearly McCoy if and only if so is \(Q\).