Over a commutative ring, $AB = I_n$ and $BA = I_m$ implies that $m = n$.
43990
R is a commutative ring with $1\ne 0$ and $R^m \cong R^n$, then $m=n$
56097
If $A$ and $AB-BA$ commute, show that $AB-BA$ is nilpotent
109498
Linear maps $f: M_n \longrightarrow \mathbb{R}$ such that $f(AB) = f(BA)$
15416
A weak converse of $AB=BA\implies e^Ae^B=e^Be^A$ from "Topics in Matrix Analysis" for matrices of algebraic numbers.
53915
prove that $a \equiv b \mod m$ is an equivalence relation on the integers
74559
$A$ and $B$ are ideals of a ring ${R}$ such that $A\cap B=\{0\}$. Prove that $st=0$ for every $s\in A, t\in B$.
68732
Prove that a semigroup satisfying $a^pb^q=ba$ is commutative
149519
Prove that a Boolean ring is a Bezout ring .
123044
Commutative matrices are multiples of the identity
18971
Show that a ring is commutative if it has the property that ab = ca implies b = c when $a\neq 0$
27766
Comaximal ideals in a commutative ring
114916
$\mathbb Z_{mn}$ isomorphic to $\mathbb Z_m\times\mathbb Z_n$ whenever $m$ and $n$ are coprime
133878
Ring of order $p^2$ is commutative.
13026
Prove that if AB=I_m, then rk(A)=rk(B).
3778
If the matrix rings over two Rings of the same size are isomorphic, then the scalar rings are isomorphic
141173
Are there matrices such that $AB=I$ and $BA \neq I$
75520
If $a^3 =a$ for all $a$ in a ring $R$, then $R$ is commutative.