In a ring, $1-ab$ is invertible $\implies$ $1-ba$ is invertible (from scratch)
83255
Finding inverse in non-commutative ring
90475
Coordinate-free proof of $\operatorname{Tr}(AB)=\operatorname{Tr}(BA)$?
66451
In a ring, how do we prove that a * 0 = 0?
131701
Why is $A^TA$ invertible if $A$ has independent columns?
119067
Basic Ring Theory Question involving the unit element
39576
Quotient ring being isomorphic to the initial ring
26570
Prove $\operatorname{rank}(AB) = \operatorname{rank}(B)$
75315
How to prove that homomorphism from field to ring is injective or zero?
119052
Finding invertible elements in $\mathbb{Z}/m\mathbb{Z}$
129827
Product of invertibles in $\Bbb Z_n$ [Wilson's Theorem generalization]
75117
$\mathrm{rank}(AB-BA)=1$ implies $A$ and $B$ are simultaneously triangularisable
18585
Showing $A-I$ is invertible, when $A$ is a skew-symmetric matrix
143051
Inverting noninvertible elements in a bigger ring.
133482
$AB-BA$ is a nilpotent matrix if it commutes with $A$
63192
A chain ring with Krull dimension greater than one
56097
If $A$ and $AB-BA$ commute, show that $AB-BA$ is nilpotent
99567
Proving that an inverse ring homomorphism of an ideal is an ideal?