Definition 2

Let A be a ring and M an A-module. A maximal submodule of M is a maximal element, for the inclusion, of the set of proper submodules of M.

A maximal submodule of $A_s$ is simply a left maximal ideal of A.

Let N be a submodule of M. The submodules of $M/N$ are of the form $P/N$, where P is a submodule of M containing N (I, §4, No. 6, p. 41, Theorem 4). Consequently, N is a maximal submodule of M if and only if the module $M/N$ is simple.