Commutative Ring Theory
《Commutative Ring Theory》 · 第 94 页 · Section 13「Graded rings」
In particular if $R_0$ is an Artinian ring, then $\ell(M_n)<\infty$, where $\ell$ denotes the length of an $R_0$-module. In this case we define the Hilbert series $P(M,t)$ of $M$ by the formula $P(M,t)=\sum_{n\geq0}\ell(M_n)t^n$.