Closure properties of

Let $\mathcal{C}$ be a class of modules and $\mathcal L = \varinjlim \mathcal C$theclassofalldirectlimitsofmodulesfrom$\mathcal C$\mathrm{.}Theclass$\mathcal L$iswellunderstoodwhen$\mathcal C$ consists of finitely presented modules: $\mathcal{L}$ then enjoys various closure properties. We study the closure properties of $\mathcal{L}$ in the general case when $\mathcal C \subseteq $\mathrm{M}\mathrm{o}\mathrm{d}-$R$ is arbitrary. Then we concentrate on two important particular cases, when $\mathcal{C}=\mathrm{add}M$ and $\mathcal C = \operatorname{Add} M$,foranarbitrarymodule$M$. In the first case, we prove that $\varinjlim \operatorname{add} M = { N \in $\mathrm{M}\mathrm{o}\mathrm{d}-$ R \mid \exists F \in \mathcal F_S: N \cong F \otimes_S M }$where$S = \operatorname{End} M$,and$\mathcal F_S$ is the class of all flat right $S$-modules. In the second case, $\varinjlim \operatorname{Add} M = { \mathfrak F \odot {\mathfrak S} M \mid \mathfrak F \in \mathcal F{\mathfrak S} }$where$\mathfrak S$istheendomorphismringof$M$ endowed with the finite topology, ${\mathcal{F}}_{\mathfrak{S}}$ is the class of all right $\mathfrak{S}$-contramodules that are direct limits of direct systems of projective right $\mathfrak{S}$-contramodules, and ${\odot }_{\mathfrak{S}}$ denotes the contratensor product. For various classes of modules $\mathcal{D}$, we show that if $M \in \mathcal D$then$\varinjlim \operatorname{add} M = \varinjlim \operatorname{Add} M$(e\mathrm{.}g\mathrm{.},when$\mathcal D$ consists of pure projective modules), but the equality for an arbitrary module $M$ remains open. Finally, we deal with the question of whether $\mathrm{}\mathrm{Add}M=\widetilde{\mathrm{Add}M}$ where $\widetilde{\mathrm{Add}M}$ is the class of all pure epimorphic images of direct sums of copies of a module $M$. We show that the answer is positive in several particular cases, but it is negative in general.