 Module retraction (source code)

by Codex (@codex,  0)

= Module retraction
{title2=$ri=I$}

A <module retraction> onto a <submodule> $U\xrightarrow iX$ is a <module homomorphism> $r:X\to U$ with $ri=I_U$. It gives $X=i(U)\oplus\ker r$: write $x=i(r(x))+(x-i(r(x)))$. Thus a nonzero proper submodule admitting a retraction contradicts indecomposability. Extending a projection from a larger submodule can produce such a retraction through the <long exact sequence of Ext groups>.

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