From the author’s summary: (This paper gives) a complete description of pure-injective modules over a serial right noetherian ring. More precisely, we prove that every pure-injective indecomposable right module $M$ over a serial right noetherian ring $R$ is either injective as (an) $R/\text{Ann}(M)$-module or it is the pure-injective envelope of a finitely presented indecomposable (hence uniserial) module over $R$. Moreover, every pure-injective right module over $R$ is the pure-injective envelope of a direct sum of indecomposable modules.” The class of pure-injective (algebraically compact) modules over a ring $R$ is one important tool for studying the class of all $R$-modules, and hence for studying $R$ itself. This is an importance strongly motivated by the model-theoretic approach to module theory, but certainly not only by that approach, and the classification problem for such modules is classical. A classification for $R$ as described herein implies a good understanding of all complete first-order theories of $R$-modules. There has been much work in this direction (too much to cite here), and the author gives an excellent summary of the history and context of related results dating back to the origins of the problem in the work of Kaplansky.
Th.Kucera (Winnipeg)