Module types (Q1382726)

The saturated classes of modules are studied and their applications to the direct sum decomposition of modules are shown. Known results on decomposition of torsion free (nonsingular) modules are extended to arbitrary modules including torsion and mixed. A saturated class of modules \(\Delta\subseteq\text{Mod-}R\) is a class of modules closed under isomorphic copies, submodules, direct sums, and injective hulls. The totality \(\Sigma(R)\) of all saturated classes of \(\text{Mod-}R\) is a set with the lattice operations \(\wedge\) and \(\vee\) where \(\wedge\Gamma=\inf\Gamma=\cap\Gamma\) and \(\vee\Gamma=\sup\Gamma=\langle\Gamma\rangle\) is the saturated class generated by \(\Gamma\subseteq\Sigma(R)\). Moreover, \(\Sigma(R)\) is a complete Boolean lattice with \(1=\text{Mod-}R\), \(0=\{(0)\}\) and each \(\Delta\in\Sigma(R)\) possesses the complement \(c\Delta\in\Sigma(R)\). Explicit descriptions of the saturated classes \(\langle\Gamma\rangle\) and \(c\Delta\) are given. If \(\Gamma\subseteq\Sigma(R)\) is a pairwise disjoint set with \(\vee\Gamma=1\in\Sigma(R)\), then for any \(R\)-module \(M\) there is an essential submodule \(\bigoplus_{\gamma\in\Gamma} M_{(\gamma)}\subseteq M\) with \(M_{(\gamma)}\in\gamma\in\Gamma\), therefore \(\widehat M=E(\bigoplus_{\gamma\in\Gamma}\widehat M_{(\gamma)})\). In the general case \(M_{(\gamma)}\) is not unique, however \(\widehat M_{(\gamma)}\) is unique up to perspectivity. A general method of building saturated classes is found and the definitions of continuous (C), discrete (D), molecular (A), continuous molecular (CA) and bottomless (B) modules are generalized for the not necessarily torsion free case. The extended definitions give again saturated classes. Parts of the older theory of types I, II and III are special cases of the more general theory of saturated classes. The mapping \(\Sigma\colon{\mathcal A}\to{\mathcal B}\) (\(R\mapsto \Sigma(R)\)) is a contravariant functor between the category of rings and the category of Boolean lattices, where every ring homomorphism \(\varphi\colon R\to S\) implies a monic lattice homomorphism \(\varphi^*\colon\Sigma(S)\to\Sigma(R)\) which preserves arbitrary infima and suprema. A universal saturated class is a function \(\Delta\) which assigns to every ring \(R\) a saturated class of \(R\)-modules \(\Delta(R)\) with coherence conditions (\(\varphi^*(\Delta(S))\subseteq\Delta(R)\), \(\varphi^*(c\Delta(S))\subseteq c\Delta(R)\)). The functions I, II, III, C, D, B, A and CA are universal saturated classes. The direct sum decomposition property of the latter classes is shown to be a special case of a general phenomenon of universal saturated classes. Let \(\{\Delta_i\mid i\in I\}\) be a family of universal saturated classes such that \(\{i\in I\mid\Delta_i(R)\neq 0\}\) is a set and \(\{\Delta_i(R)\mid i\in I\}\subseteq\Sigma(R)\) is a pairwise orthogonal subset with \(\sup\{\Delta_i(R)\mid i\in I\}=1\in\Sigma(R)\). Then for any ring \(R\) and for any module \(M_R\) there is an essential inclusion \(\bigoplus_{i\in I}M_i\subseteq M\), \(M_i\in\Delta_i(R)\), and \(\widehat M=E(\bigoplus_{i\in I}\widehat M_i)\), \(\widehat M_i\in\Delta_i(R)\). It is shown that there is a class of pairwise disjoint universal saturated classes one for each cardinal number. The main results are illustrated by numerous applications and various examples.