Title: 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.