Counting submodules of a module over a Noetherian commutative ring (Q2312870)

It is known that the abelian groups whose set of subgroups is countable are exactly those that are minimax and do not admit \(C_{p^{\infty}}^{2}\) as a subquotient for any prime \(P\) ([\textit{D. L. Boyer}, Proc. Am. Math. Soc. 7, 565--570 (1956; Zbl 0071.02402)]). The paper under review studies the generalization of this result to modules over associative, unital, commutative rings. In the case of a module \(M\) over a countable noetherian ring \(A\), it is shown that the set of submodules of \(M\) is countable if and only if \(M\) is minimax and satisfies two properties called \((L_{2})\) and \((L_{1+1})\). An \(A\)-module \(M\) is said to satisfy \((L_{2})\) if for every artinian quotient \(Q\) of \(M\), the poset of submodules of \(Q\) does not contain any chain isomorphic to the ordinal \(\omega^{2}\), while \(M\) satisfies \((L_{1+1})\) if no subquotient of \(M\) is isomorphic to the square \(P \oplus P\) of some \(A\)-module \(P\) of infinite length. The Matlis duality is used extensively in the proof of the above-mentioned main theorem. Going beyond the countable case, the number of submodules of a minimax \(A\)-module, where \(A\) is a Noetherian ring of cardinal \(\alpha\), all of whose quotient fields also have cardinal \(\alpha\), is determined. Here different results are found for the case when \(M\) is meager and when it is not. (An \(A\)-module \(M\) is called meager if \(M\) does not admit \(S^{2}\) as a subquotient for any simple \(A\)-module \(S\).) As a corollary to this counting theorem, an interesting characterization of uniserial modules is found. Meager modules as such are also studied and various properties and characterizations are found.