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

## TL;DR

This paper explores properties of annihilators in graded rings, showing that non-trivial right ideals have homogeneous annihilators in certain graded structures, extending previous results to more general settings.

## Contribution

It generalizes McCoy's result by demonstrating that in $
abla$-graded rings, right annihilators contain homogeneous elements, and under certain conditions, degree 0 annihilators can be found.

## Key findings

- Non-trivial right ideals in $
abla$-graded rings have homogeneous annihilators.
- In subrings of $bZ^k$-graded rings with non-annihilation properties, degree 0 annihilators exist.
- Extension of annihilator properties from polynomial rings to more general 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.