Pure projectives and injectives.
A module over a ring \(R\) is pure projective or pure injective if it has the (category-theoretic) injective or projective property relative to all pure short exact sequences. Pure exactness, pure injectivity, and indeed injectivity can all be characterized by conditions in terms of the solvability of systems of linear equations. By putting restrictions on the cardinality of the set of variables or on the cardinality of the set of equations, the author obtains a large class of variations on the definitions of pure injectivity and pure projectivity. (For simplicity in this brief review, we avoid the technical details of these variations and just refer to \(\mathcal P\)-purity, etc., and suppress reference to restrictions on the range of cardinalities considered.) Some well-known properties generalize in a natural way to this new context: for instance, a module is \(\mathcal P\)-pure projective if and only if it is a direct summand of a direct sum of \(\mathcal P\)-presented modules; and, in a slightly different context, a module is \(\mathcal P\)-pure injective if and only if it is a direct summand in every module in which it is contained as a \(\mathcal P\)-pure submodule. There is an interesting open problem at the end, namely the existence of \(\mathcal P\)-pure injective hulls. Likely a solution involves some additional set theory. Purity, pure injectivity and variations on them have been of central importance in the model theory of modules. The characterizations of these natural algebraic concepts by properties of solutions of systems of linear equations relates these concepts to the model-theoretic property of saturation, a central tool of the subject area. The variations of pure injectivity presented in this paper can all be viewed as weak saturation conditions, and further work on this subject might profit from exploring this connection in more detail. In this context there has been a fair amount of work on variations of concepts of purity and of pure projectivity which should be integrated with the work presented here. See especially various articles by Ph.\ Rothmaler et al.: \textit{G. Puninski}, \textit{M. Prest} and \textit{Ph. Rothmaler}, Commun. Algebra 27, No. 5, 2127--2162 (1999; Zbl 0943.16004); \textit{T. Kucera} and \textit{Ph. Rothmaler}, J. Symb. Log. 65, No. 1, 103--110 (2000; Zbl 0949.03032); \textit{Ph. Rothmaler}, Algebra Log. Appl. 9, 445--469 (1997; Zbl 0931.03055); \textit{Ph. Rothmaler}, Ann. Pure Appl. Logic 88, No. 2--3, 227--239 (1997; Zbl 0916.16004).
scientific article; zbMATH DE number 2012075
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