= Noncommutative Hilbert basis theorem
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 $F_n=\sum_{i\leq n}Ax^i=\sum_{i\leq n}x^iA$ and, for each <right ideal> $I$, the ascending <right ideals> $L_n=\{a:ax^n\in I+F_{n-1}\}$. After these stabilize, finitely many lifts of generators for $L_0,\ldots,L_N$ generate $I$ by induction on degree. Neither unique normal forms nor an <automorphism> moving coefficients is required.
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