In this paper we characterize weak multiplication modules.

Let R be the pullback, in the sense of Levy [J. Algebra 71 (1981)], of two local Dedekind domains. We classify all those indecomposable weak multiplication R-modules M with finite-dimensional top, that is, such that M/Rad(R)M is finite-dimensional over R/Rad(R). We also establish a connection between the weak multiplication modules and the pure-injective modules over such domains.

A lattice-ordered ring $ℝ$ is called an OIRI-ring if each of its order ideals is a ring ideal. Generalizing earlier work of Basly and Triki, OIRI-rings are characterized as those $f$-rings $ℝ$ such that $ℝ/𝕀$ is contained in an $f$-ring with an identity element that is a strong order unit for some nil $l$-ideal $𝕀$ of $ℝ$. In particular, if $P\left(ℝ\right)$ denotes the set of nilpotent elements of the $f$-ring $ℝ$, then $ℝ$ is an OIRI-ring if and only if $ℝ/P\left(ℝ\right)$ is contained in an $f$-ring with an identity element that is a strong order unit....