A ring is <Left Noetherian ring>[left Noetherian] when its left ideals satisfy the ascending-chain condition, equivalently when every left ideal is finitely generated; right Noetherian is defined analogously.

Filter $S$ by word degree in $x$. The equality $R+Rx=R+xR$ lets every coefficient move past one $x$ at the cost of lower-degree terms, so
$$
F_nS=R+Rx+\cdots+Rx^n.
$$
For a left ideal $I$, the leading coefficients in degree at most $n$ form an ascending chain of left ideals of $R$. Since $R$ is left Noetherian, this chain stabilizes and each term is finitely generated. Lift finitely many generators through the finitely many degrees before stabilization. Division by their leading terms reduces every element of $I$ to lower degree, and induction shows that these lifts generate $I$. Thus $S$ is left Noetherian.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 139 / 1 / a / Solution

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{wiki=Noetherian_ring}

A ring is left Noetherian when every ascending chain of left ideals stabilizes, equivalently when every left ideal is finitely generated.


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