Purity relative to classes of finitely presented modules. (Q2847697)

This paper explores in detail the purities determined by classes of finitely presented unital modules over associative rings. Section 2 of the work offers many characterizations and properties of \(S\)-purity, \(S\)-pure-injectivity, and \(S\)-pure-projectivity. In section 3 the author compares \(S\)-purity and \(T\)-purity for arbitrary classes \(S\) and \(T\) of finitely presented left \(R\)-modules. Section 4 looks at \((n,m)\)-purity over semiperfect rings. Section 5 examines purity over finite-dimensional algebras and includes a description of the pure-injectives in the case of tame hereditary algebras. The final section, 6, includes a condition on a left \(R\)-module \(M\) such that every \(S\)-pure submodule of \(M\) is a direct summand and shows that such a module is a direct sum of indecomposable submodules. As a consequence the author finds a characterization of rings over which every indecomposable left \(R\)-module is \(S\)-pure-projective.NEWLINENEWLINE The results of this paper are part of the author's doctoral thesis.