Anti-homomorphisms between module lattices

Abstract

We examine the properties of certain mappings between the lattice $\mathcal{L}(R)$ of ideals of a commutative ring $R$ and the lattice $\mathcal{L}(_RM)$ of submodules of an $R$-module $M$, in particular considering when these mappings are lattice anti-homomorphisms. The mappings in question are the mapping $\alpha : \mathcal{L}(R) \rightarrow \mathcal{L}(_RM)$ defined by setting for each ideal $B$ of $R$, $\alpha(B)$ to be the submodule of $M$ consisting of all elements $m$ in $M$ with $Bm = 0$ and the mapping $\beta : \mathcal{L}(_RM) \rightarrow \mathcal{L}(R)$ defined by $\beta(N)$ is the annihilator in $R$ of $N$, for each submodule $N$ of $M$.