Absolutely pure and flat modules and ``indiscrete'' rings
The authors investigate the rings over which all nonzero modules are indistinguishable by the elementary statements in the language of modules. They prove that infinite indiscrete rings are rings of this type (they define a ring to be indiscrete if its Ziegler spectrum carries the indiscrete topology). They prove that a ring is indiscrete if and only if it is simple and all its (left as well as right) finitely presented modules are absolutely pure. Further, if \(R\) is an infinite simple ring and if all its finitely presented left modules are absolutely pure and \(R\), as a right \(R\)-module, is absolutely pure, then \(R\) is indiscrete. They derive some sufficient conditions for a simple ring to be indiscrete in terms of left and right Ziegler spectra of \(R\). They prove that an indiscrete ring \(R\) is regular if and only if the left (right) annihilator of every element of \(R\) is finitely generated. If \(R\) is an infinite, simple, left and right absolutely pure ring, then all injective left \(R\)-modules are elementarily equivalent. They also prove that if \(T\) is a complete theory of right \(R\)-modules, then every model of \(T\) is absolutely pure if and only if every model of \(DT\) is flat. They close the paper with the result that if \(R\) is an indiscrete ring with maximal left quotient ring, then \(R\) is regular if and only if the pure-injective hull of the left \(R\)-module \(R\) is flat.
scientific article; zbMATH DE number 783868