Krull's intersection theorem in case of small submodules
Let \(R\) denote a commutative Noetherian ring and \(M\) an \(R\)-module. A submodule \(U\) of \(M\) is called small in case \(U + X \subsetneqq M\) for each submodule \(X \subsetneqq M\). It is common that the sum of all small submodules of \(M\) is the Jacobson radical \(\text{rad} M\) of \(M\) and that finitely generated submodules of \(\text{rad} M\) are small. In general, small submodules need not be finitely generated. However, it is shown in the present article that under reasonable assumptions small submodules share an important finiteness property with the finitely generated ones, namely the Krull intersection theorem. For instance, the latter holds for small submodules of torsion-free modules in case \(R\) is an integral domain. A consequence of this result is that over a local ring flat small submodules are finitely generated. Two applications of the finiteness conditions for small submodules are given. First, they are used for a very explicit description of the set of coassociated prime ideals of a flat module \(M\). (These are the prime ideals of \(R\) which are annihilators of artinian factor modules of \(M\).) -- Second, conditions are given under which rings between \(R\) and its total quotient ring \(K\) are small submodules of the \(R\)-module \(K\). It is shown that the integral closure \(R'\) of \(R\) in \(K\) has this property, iff each maximal ideal of \(R\) is regular, and conversely that a ring \(R \subset A \subset K\) must be contained in \(R'\) provided that it is small in \(K\) and \(A/{\mathfrak m} A\) is a finitely generated \(R\)-module for each maximal ideal \({\mathfrak m}\) of \(R\).
coassociated prime ideals of a flat module
scientific article; zbMATH DE number 563363