Dedekind-finite ring

Remarks

Example: for a field $F$ and $n \ge 1$, the matrix ring $M_n(F)$ is Dedekind-finite. Indeed, if $AB = I$, then $\det A \det B = 1$, so $A$ is invertible, and $B = A^{-1}(AB) = A^{-1}$, hence $BA = I$.

Non-example: let $V$ be the vector space of sequences $(x_1, x_2, \dots)$ over a field and $R = \End(V)$. Let $a(x_1, x_2, \dots) = (x_2, x_3, \dots)$ and $b(x_1, x_2, \dots) = (0, x_1, x_2, \dots)$. Then $ab = 1$, but $ba(x_1, x_2, \dots) = (0, x_2, x_3, \dots)$, so $ba \ne 1$.

Other names are directly finite and von Neumann finite. Equivalently, $R$ is not isomorphic as a left $R$-module to $R \oplus N$ for any nonzero module $N$. Every commutative ring, every finite ring, and every left Noetherian ring is Dedekind-finite.