Hilbert–Samuel function

In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel, of a nonzero finitely generated module ${\displaystyle M}$ over a commutative Noetherian local ring ${\displaystyle A}$ and a primary ideal ${\displaystyle I}$ of ${\displaystyle A}$ is the map ${\displaystyle \chi _{M}^{I}:\mathbb {N} \rightarrow \mathbb {N} }$ such that, for all ${\displaystyle n\in \mathbb {N} }$,

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