The covers of a Noetherian module (Q1919920)

Let \(R\) be a commutative ring and \(A\) an \(R\)-module. A cover of \(A\) is defined to be a subset \(T\) of \(\text{Max} (R)\) satisfying that for any \(x \in A\), \(x \neq 0\), there is \(M \in T\) such that \(0:{}_R x\subseteq M\). If we denote by \(J\) the intersection of all the maximal ideals contained in \(T\) and suppose that \(A \neq 0\) is finitely generated, then we have \(JA \neq A\). This generalises Nakayama's lemma; if, in addition, \(R\) is Noetherian, then \(\bigcap^\infty_{n = 1} J^n A = 0\). This is a generalization of a well-known result. A key observation for the converse is that, in the case that \(R\) is Noetherian and \(A\) is finitely generated, there is a cover \(T\) of \(A\) which is itself a finite set. From this we have the following result: Let \(R\) be a Noetherian ring. Then there is a finite number of maximal ideals \(M_1, \dots, M_s\) of \(R\) such that \(\bigcap^\infty_{n = 1} J^n = 0\), where \(J = \bigcap^s_{i = 1} M_i\). This generalises Krull's theorem for Jacobson radicals. Using this result we can embed the Noetherian \(R\) in the \(J\)-adic completion \(\widehat R\) of \(R\), which is a complete semilocal Noetherian ring; besides, if \(R\) is a CM ring, then \(\widehat R\) is a CM ring. We also use the covers to deal with the maximal component of a finitely generated module over a Noetherian ring, which was introduced by Matlis.