 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 $F_n={\sum }_{i\le n}A{x}^i={\sum }_{i\le n}{x}^{i}A$ 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.