Mappings between the lattices of saturated submodules with respect to a prime ideal
Mappings between the lattices of saturated submodules with respect to a prime ideal
Let $\mathfrak{S}_p(_RM)$ be the lattice of all saturated submodules of an $R$-module $M$ with respect to a prime ideal $p$ of a commutative ring $R$. We examine the properties of the mappings $\eta:\mathfrak{S}_p(_RR)\rightarrow \mathfrak{S}_p(_RM)$ defined by $\eta(I)=S_p(IM)$ and $\theta:\mathfrak{S}_p(_RM)\rightarrow \mathfrak{S}_p(_RR)$ defined by $\theta(N)=(N:M)$, in particular considering when these mappings are …