Proposition 10

a) The B-module V is Artinian (resp. Noetherian, indecomposable, simple, finitely generated) if and only if M is.

b) The B-module V has finite length if and only if the A-module M has finite length, and we then have long$_B(V) =$ long$_A(M)$.

c) The B-module V is semisimple (resp. isotypical) if and only if the A-module M is semisimple (resp. isotypical). If this is the case, then we have long$_B(V) =$ long$_A(M)$.

Assertion a) follows from the second property listed in Table II.

Assertion b) follows from Proposition 8, d).

The module V is semisimple if and only if it is equal to its socle $\mathscr{S}(V)$; it is isotypical if and only if there exists a simple submodule S of V such that $V = V_S$. Assertion c) therefore follows from Propositions 7, c), 8, c), and 9 (VIII, p. 106 and 109).