Remarks on relatively flat modules
Let \(R\) be a ring with identity element, and let \(I\) be an idempotent ideal of \(R\). Then \(\mathfrak F = \{M_R | MI = 0\}\) is a \(TTF\) class of unital right \(R\)-modules in the sense of \textit{J. P. Jans} [Pac. J. Math. 15, 1249--1259 (1965; Zbl 0142.28002)]. A left \(R\)-module \(M\) is relatively flat if, for an exact sequence \(0 \to A \to B \to C \to 0\) of right \(R\)-modules such that \(C \in \mathfrak F\), the sequence \(0 \to A \otimes_R M \to B \otimes _R M\) is exact. An exact sequence \(0 \to L \to X \to M \to 0\) of left \(R\)-modules is \(I\)-pure if, for each \(A \in \mathfrak F\), the sequence \(0 \to A\otimes_R L \to A\otimes_R X\) is exact. A submodule \(U\) of a module \(V\) is \(I\)-pure if the sequence \(0 \to U \to V \to V/U \to 0\) is \(I\)-pure. This paper proves some basic properties of \(I\)-flat modules and \(I\)-pure submodules. The main result is the following element-wise characterization of \(I\)-purity. Proposition. If \(I\) is finitely generated as a right ideal and if \(U\) is a submodule of \(V\), then the following two conditions are equivalent. (1) \(U\) is an \(I\)-pure submodule of \(V\). (2) Let \(v_1,\dots v_m \in V\), \(u_1,\dots u_n \in U\), and \(d_{ij} \in R (i=1,\dots ,n; j=1,\dots ,m)\), where \(\{d_{ij}\}\) satisfies the following condition: for any \(x_1,\dots x_n \in I\), there exist \(r_1,\dots r_m \in R\) such that \(x_i = \sum_{j} d_{ij}r_j (i =1,\dots ,n)\). If \(u_1 = \sum_{j} d_{ij}r_j\), then there exist \(u'_1,\dots u'_m \in U\) such that \(u_i = \sum_{j} d_{ij}u'_j (i =1,\dots ,n)\).
scientific article; zbMATH DE number 3557994