# Pure-injective modules

Prest, Mike (2008) Pure-injective modules. [MIMS Preprint]

The pure-injective $R$-modules are defined easily enough: as those modules which are injective over all pure embeddings, where an embedding $A\rightarrow B$ is said to be pure if every finite system of $R$-linear equations with constants from $A$ and a solution in $B$ has a solution in $A.$ But the definition itself gives no indication of the rich theory around purity and pure-injectivity. The purpose of this survey is to present and illustrate the definitions and a number of the results around pure-injective modules.