Corollary

Every submodule of a monogenous A-module E is isomorphic to a quotient module b/a where a and b are two left-ideals of A such that a ⊂ b. Every quotient module of a monogenous A-module is monogenous.

The second assertion is immediate and the first follows from Proposition 22 and I, § 4, no. 6, Theorem 4.

Note on the other hand that a submodule of a monogenous module is not necessarily monogenous. For example, if A is a commutative ring in which there exist non-principal ideals (VII, § 1, no. 1), these ideals are non-monogenous submodules of the monogenous A-module A.

It follows from the definitions that a submodule of an A-module E generated by a family $(a_i)$ of elements of E is the sum of the monogenous submodules

Aa_i of E; for (a_i) to be a basis of E, it is necessary and sufficient that each of the a_i be a free element of E and that the sum of the Aa_i be direct.