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A Noetherian module satisfies the ascending chain condition on its submodules; equivalently every submodule is a finitely generated module. If the latter condition holds, the union of an ascending chain has a finite generating set lying in one member, so the chain stabilizes. Conversely, a submodule with no finite generating set allows successive choices outside the submodule generated so far, producing a strictly ascending chain. A Noetherian ring is a ring that is Noetherian as a module over itself, equivalently one whose ideals are all finitely generated.

The formal power series ring consists of sequences written ${\sum }_{i\ge 0}{a}_i{x}^i$, with coefficientwise addition and the Cauchy product:Every coefficient uses a finite sum; no analytic convergence is involved. Evaluation at zero gives $A[[x]]/(x)\cong A$. A quotient ring of a Noetherian ring is Noetherian, because its ascending chains of ideals lift to such chains upstairs. Thus one direction of Noetherianity of a formal power series ring is immediate.

For the other direction suppose $A$ is Noetherian, and let $I\subseteq A[[x]]$ be an ideal. Define the coefficient ideals of a formal power series idealEach is an ideal, and multiplication by $x$ gives $I_r\subseteq I_{r+1}$. Choose $N$ with $I_N=I_{N+1}=\cdots$. For $0\le r\le N$, choose generators $a_{r1},\dots ,a_{r{m}_r}$ for $I_r$, and choose series $g_{rj}\in I\cap x^{r}A[[x]]$ whose $x^r$ coefficients are those generators. Empty generating sets are allowed for zero ideals.

These finitely many series generate $I$. To prove this without assuming that $I$ is closed, start with $g\in I$ and cancel its coefficients successively. If the residual $h_r$ lies in $I\cap x^{r}A[[x]]$, its $x^r$ coefficient lies in $I_r$. Put $s=\min (r,N)$; since $I_r=I_s$ when $r\ge N$, write that coefficient as ${\sum }_j{c}_{rj}{a}_{sj}$ and subtract ${\sum }_j{c}_{rj}{x}^{r-s}{g}_{sj}$. The new residual belongs to $I\cap x^{r+1}A[[x]]$.

For $s<N$, the generators $g_{sj}$ are used only at the single step $r=s$. For $s=N$, their accumulated multipliers are the genuine formal power series ${\sum }_{r\ge N}{c}_{rj}{x}^{r-N}$. Thus the construction yields the exact coefficientwise identityEvery coefficient of the residual vanishes after finitely many steps. This is an ordinary finite generating expression in $A[[x]]$, not merely an approximation by elements of a smaller ideal. Therefore every ideal is finitely generated, provingFinally, for the surjective module endomorphism $f$, the kernels $\ker f\subseteq \ker f^2\subseteq \cdots$ stabilize because $M$ is a Noetherian module. Take $N$ with $\ker f^N=\ker f^{N+1}$. If $f(m)=0$, surjectivity of $f^N$ gives $u$ with $f^N(u)=m$. Then $f^{N+1}(u)=0$, so $u\in \ker f^N$ and $m=0$. Hence $f$ is injective as well as surjective; its inverse is automatically $A$-linear. This proves $f$ is an isomorphism, the statement that Noetherian modules are Hopfian.