Right Noetherian ring

A ring is right Noetherian when every ascending chain of right ideals stabilizes.

Suppose $A\subseteq B$ are unital rings and $B$ is a finitely generated module over $A$ on the right. If $A$ is a right Noetherian ring, then $B_A$ is a Noetherian module. Every right ideal of $B$ is an $A$-submodule, hence has finitely many generators over $A$; these also generate it over $B$. Thus $B$ is a right Noetherian ring. The same argument works on the left under the corresponding hypotheses. Finite-index group ring extensions give an important example, using coset representatives as a finite module basis.

A noncommutative domain whose right ideals satisfy the ascending chain condition is a right Noetherian domain. Its right regular module is a uniform module, so any two nonzero principal right ideals intersect. Thus its nonzero elements satisfy the right Ore condition and it embeds in a division ring by Ore localization.

If an algebra $B$ is generated by a subalgebra $A$ that is a right Noetherian ring and one element $x$, and $A+xA=A+Ax$, then $B$ is a right Noetherian ring. The proof uses $F_n={\sum }_{i\le n}A{x}^i={\sum }_{i\le n}{x}^{i}A$ and, for each right ideal $I$, the ascending right ideals $L_n=\{a:a{x}^n\in I+F_{n-1}\}$. After these stabilize, finitely many lifts of generators for $L_0,\dots ,L_N$ generate $I$ by induction on degree. Neither unique normal forms nor an automorphism moving coefficients is required.

For a right ideal $I\subseteq R[t]$, let $L_d$ be the set of degree-$d$ coefficients of its elements of degree at most $d$. These are right ideals of $R$, and multiplication by the central variable makes $L_d\subseteq L_{d+1}$. They stabilize, and every $L_d$ before stabilization is finitely generated. Choose polynomial lifts of those generators. Subtracting a suitable right linear combination of the lifts, shifted by powers of $t$, cancels the leading coefficient of any element of $I$. Induction on degree proves that the finitely many chosen lifts generate $I$.

// Destination: noncommutative-algebra.bigb

For an automorphism $\sigma$ of a left Noetherian ring $A$, the skew polynomial ring of an automorphism $A[t;\sigma ]$ is left Noetherian. For a left ideal $L$, the normalized leading-coefficient ideals $I_n=\{{\sigma }^{-n}(a_n):{\sum }_{j=0}^n{a}_j{t}^j\in L\}$ form an ascending chain of left ideals of $A$. Choose finite generating lifts through the stabilization degree. Every element of $L$ has its leading coefficient cancelled by a suitable left multiple of one of these lifts, and induction on degree proves finite generation. Passing to the skew Laurent polynomial ring preserves the property by clearing negative powers. The right-sided statement follows by applying the argument to the opposite ring.