We first determine the domination number for the zero-divisor graph of the product of two commutative rings with 1. We then calculate the domination number for the zero-divisor graph of any commutative artinian ring. Finally, we extend some of the results to non-commutative rings in which an element is a left zero-divisor if and only if it is a right zero-divisor.

Suppose that $G$ is an abelian group, $A\,\subset \,G$ is finite with $\left| A\,+\,A \right|\,\le \,K\left| A \right|$ and $\eta \,\in \,(0,\,1]$ is a parameter. Our main result is that there is a set $L$ such that

$$\left| A\,\cap \,\text{Span}\left( L \right) \right|\ge {{K}^{-{{O}_{n}}\left( 1 \right)}}\left| A \right|\,\,\,\,\text{and}\,\,\,\,\,\left| L \right|=O\left( {{K}^{n}}\log \left| A \right| \right).$$

We include an application of this result to a generalisation of the Roth-Meshulam theorem due to Liu and Spencer

We prove that a finitely generated group contains a sequence of non-trivial elements that converge to the identity in every compact homomorphic image if and only if the group is not virtually abelian. As a consequence of the methods used, we show that a finitely generated group satisfies Chu duality if and only if it is virtually abelian.

Following ideas used by Drewnowski and Wilansky we prove that if $I$ is an infinite dimensional and infinite codimensional closed ideal in a complete metrizable locally solid Riesz space and $I$ does not contain any order copy of ${{\mathbb{R}}^{\mathbb{N}}}$ then there exists a closed, separable, discrete Riesz subspace $G$ such that the topology induced on $G$ is Lebesgue, $I\,\bigcap \,G\,=\,\left\{ 0 \right\}$, and $I\,+\,G$ is not closed.

Let $G$ be an infinite discrete group and let $\beta G$ be the Stone-Čech compactification of $G$. We take the points of $\beta G$ to be the ultrafilters on $G$, identifying the principal ultrafilters with the points of $G$. The set $U\left( G \right)$ of uniform ultrafilters on $G$ is a closed two-sided ideal of $\beta G$. For every $p\,\in \,U\left( G \right)$, define ${{I}_{p}}\,\subseteq \,\beta G$ by ${{I}_{p}}\,=\,{{\bigcap }_{A\in p}}\text{cl}\left( GU\left( A \right) \right)$, where $U\left( A \right)\,=\,\left\{ p\,\in \,U\left( G \right)\,:\,A\in \,p \right\}$. We show that if $\left| G \right|$ is a regular cardinal, then $\left\{ {{I}_{p}}\,:\,p\,\in \,U\left( G \right) \right\}$ is the finest decomposition of $U\left( G \right)$ into closed left ideals of $\beta G$ such that the corresponding quotient space of $U\left( G \right)$ is Hausdorff.