if $a,b$ are elements of a unital algebra $A,$ then $1-ab$ is invertible if and only if $1-ba$ is invertible.
64077
$I_m - AB$ is invertible if and only if $I_n - BA$ is invertible.
8787
In a ring $(A,+, \cdot)$ if $aba = a$ then $bab = b$ and all non zero elements in $A$ are invertible.
37182
If an element has a unique right inverse, is it invertible?
119052
Finding invertible elements in $\mathbb{Z}/m\mathbb{Z}$
108781
$A$ and $B$ are bounded linear operators from the normed linear space $X$ to itself. If $AB$ is invertible are $A$ and $B$ invertible?
157784
Prove that if $\gcd(ab,c)=1$, then $\gcd(a,c)=1$.
64108
how to prove $a+b-ab \le 1$ if $a,b \in [0,1]$?
55018
Prove that $A^*A + I$ is invertible
21726
Show that if $(b,c)=1$, then $(a,bc)=(a,b)(a,c)$.
145876
$AB-BA=A$ implies $A$ is singular and $A$ is nilpotent.
10348
Prove that if the $\gcd(a,b)=1$, then $\gcd(a+b,ab)=1$
84422
$AB=BA$, $AC=CA$ we need to show $BC=CB$
105921
Why do we have $ab=ba$?
50266
In the following figure prove that: $AK||BC$
1448
$x^a - 1$ divides $x^b - 1$ if and only if $a$ divides $b$
78736
Check my proof that $(ab)^{-1} = b^{-1} a^{-1}$
16023
Are associates unit multiples in a commutative ring with $1$?