Annihilators of polynomials

Abstract

In 1942, McCoy proved that if a polynomial f(x) over a commutative ring R has a nonzero annihilator in then it has a nonzero annihilator in R also. Later in 1957 McCoy proved that for R an arbitrary ring, if is annihilated on the right by some nonzero polynomial in then is also annihilated on the right by some nonzero element of R. Generalizing these results of McCoy we prove that a polynomial f(x) over an arbitrary ring R has a nonzero right annihilator in R if and only if there exists a nonzero polynomial g(x) in such that for every c in the multiplicative monoid generated by the coefficients of f(x). We also show that taking c to be identity and coefficients of f(x) does not suffice. We also give a new proof of Nielsen’s result that every reversible ring is right McCoy. We prove that an exchange right linearly McCoy ring is reduced modulo the Jacobson radical. We also prove that is a McCoy ring, where S and T are local rings with nonzero Jacobson radicals in which product of two non-units is zero and is a bimodule such that and MJ(T) = 0. This leads to a large class of exchange non-abelian McCoy rings.