Annihilators of polynomials

Abstract

In 1942, McCoy proved that if a polynomial f(x) over a commutative ring R has a nonzero annihilator in $R[x],$ then it has a nonzero annihilator in R also. Later in 1957 McCoy proved that for $f(x)\in R[x],$ R an arbitrary ring, if $f(x)R[x]$ is annihilated on the right by some nonzero polynomial in $R[x],$ then $f(x)R[x]$ 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 $R[x]$ such that $f(x)\mathit{cg}(x)=0$ 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 $R=\left(\begin{array}{cc}S& M\\ 0& T\end{array}\right)$ 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 ${}_S{M}_T$ is a bimodule such that $J(S)M=0$ and MJ(T) = 0. This leads to a large class of exchange non-abelian McCoy rings.

Keywords:

• Annihilators
• McCoy rings
• linearly McCoy rings
• reversible rings
• Dedekind finite rings
• Abelian rings

MSC:

• Primary: 16U80
• Secondary 16U99
• 16S36

Notes

1 A ring whose all idempotents are central is called abelian.

2 A ring R is called semi-commutative if ab = 0 implies aRa = 0 for every $a,b\in R.$