Proposition 7

a) We have $\varphi (0) = 0$ and $\varphi (V) = M$.

b) Let $(V_i)_{i\in I}$ be a family of submodules of V. We have

Submodules of M Ordered set D(M)

Zero submodule Smallest element of D(M)

Submodule M Greatest element of D(M)

$\bigcap_{i\in I}M_i$ Greatest lower bound inf$_{i\in I}M_i$

$\sum_{i\in I}M_i$ Least upper bound sup$_{i\in I}M_i$

Supplementary submodules inf(M$',M''$) $= 0$, sup(M$',M''$) $= M$

Simple submodule of M Minimal element of D(M) $-\{0\}$

Maximal submodule of M Maximal element of D(M) $-\{M\}$

Socle $\mathscr{S}(M)$ of M Least upper bound in D(M) of the set of

minimal elements of D(M)$-\{0\}$

$*$Radical $\mathfrak{R}(M)$ of M Greatest lower bound in D(M) of the set

(VIII, p. 151) of maximal elements of D(M)$-\{M\}$

Table I.

Properties of the module M Properties of D(M)

M is Noetherian. The ordered set D(M) is Noetherian (Set

Theory, III, §6, No. 5, p. 190).

M is Artinian. The set D(M), ordered by $\supset$, is

Noetherian.

M is indecomposable. We have M $\not= 0$, and there are no

two nonzero elements $M'$ and $M''$ of

D(M) satisfying inf(M$',M''$) $= 0$,

sup(M$',M''$) $= M$.

M is finitely generated. For every family $(M_i)_{i\in I}$ in D(M) with

upper bound M, there exists a finite

subset J of I such that M = sup$_{j\in I}M_j$. M is simple. Card(D(M)) = 2.

M is semisimple. M is the least upper bound, in D(M),

of the set of minimal elements of

D(M)$-\{0\}$.

Table II.

c) The B-module V is the direct sum of the family $(V_i)_{i\in I}$ of submodules if and only if M is the direct sum of the family $(\varphi (V_i))_{i\in I}$.

Let $V'$ and $V''$ be submodules of V such that $V'$ is contained in $V''$; set $M'=\varphi (V')$ and $M''=\varphi (V'')$, so that $M''$ contains $M'$. Denote by $[V',V'']$ the interval in D(V) consisting of the submodules W of V such that we have $V'\subset W\subset V''$, and define the interval $[M',M'']$ in D(M) likewise. The mapping $W\mapsto W/V'$ is an isomorphism of ordered sets from $[V',V'']$ to $D(V''/V')$; we define an isomorphism of ordered sets from $[M',M'']$ to $D(M''/M')$ likewise. Since $\varphi$ sends the interval $[V',V'']$ to $[M',M'']$, it defines an isomorphism $\varphi$ of ordered sets from $D(V''/V')$ to $D(M''/M')$. We deduce the following proposition from this and Tables I and II.