Annihilators in $\mathbb{N}^k$-graded and $\mathbb{Z}^k$-graded rings

Abstract

It has been shown by McCoy that a right ideal of a polynomial ring with several indeterminates has a non-trivial homogeneous right annihilator of degree 0 provided its right annihilator is non-trivial to begin with. In this note, it is documented that any $\mathbb{N}$-graded ring $R$ has a slightly weaker property: the right annihilator of a right ideal contains a homogeneous non-zero element, if it is non-trivial to begin with. If $R$ is a subring of a $\mathbb{Z}^k$ -graded ring $S$ satisfying a certain non-annihilation property (which is the case if $S$ is strongly graded, for example), then it is possible to find annihilators of degree 0.