Decisive dimension and other related torsion theoretic dimensions. (Q863903)

A left \(R\)-module \(M\) is called `decisive' if for every hereditary torsion theory \(\tau\), in the category of left \(R\)-modules, \(M\) is either \(\tau\)-torsion or \(\tau\)-torsion-free. In the present paper `the decisive dimension' of a module is defined using the torsion theories which are cogenerated by decisive modules. Section 2 of the paper is dedicated to the study of this dimension. The authors are concerned especially to the question ``when a left \(R\)-module has decisive dimension?''. In Section 3, to each left module \(M\), a class of torsion theories which are cogenerated by decisive modules is associated. Using all these the authors find in Section 4 new characterizations for left semiartinian rings (Corollary 4.2) and for left Artinian rings (Theorem 4.3). I want to mention that the paper contains many interesting and illuminating examples.