= Subdirect product of pairwise nonisomorphic simple modules
{title2=$W\leq\bigoplus_{i=1}^kS_i,\quad W\twoheadrightarrow S_i\implies W=\bigoplus_iS_i$}
Suppose the $S_i$ are pairwise nonisomorphic <simple modules>. Any <submodule> $W$ of their finite <direct sum> that projects onto every summand is the entire sum. Induct on $k$. Projection onto the first $k-1$ factors is onto by induction. Its <kernel> is a <submodule> of $S_k$, hence zero or all of $S_k$. If zero, $W$ is isomorphic to the first $k-1$ factors but also has quotient $S_k$, contradicting their distinct <composition factors> and the <Jordan–Hölder theorem>. Therefore the <kernel> is $S_k$, proving the assertion. Consequently, if $S_i=Ax_i$, the map $a\mapsto(ax_i)_i$ is onto and gives $A/(\bigcap_i\operatorname{Ann}_A(x_i))\cong\bigoplus_iS_i$ as left modules. The annihilators of these vectors need only be left ideals.
Back to article page
