In a finite ring $R$ with identity show that $ab =1$ implies $ba = 1$, where $a,b \in R$.
48009
Left inverse implies right inverse in a finite ring
65224
Prove that if $a^2 = a$ for all elements of the ring $R$, then $R$ has characteristic 2. Is the converse of the statement true?
75117
$\mathrm{rank}(AB-BA)=1$ implies $A$ and $B$ are simultaneously triangularisable
98331
Show that if $ab$ has finite order $n$, then $ba$ also has order $n$. - Fraleigh p. 47 6.46.
84422
$AB=BA$, $AC=CA$ we need to show $BC=CB$
99567
Proving that an inverse ring homomorphism of an ideal is an ideal?
118981
Proving $\sqrt{ab} = \sqrt a\sqrt b$
150309
Why do we assume $a+b=b+a$ in a Ring with 1. Also, is it true with Rings without 1?
128820
Prove that if $\gcd(a,b)=1$ then $\gcd(ab,c) = \gcd(a,c) \gcd(b,c)$
102742
order $a$ = 5, $a^3b = ba^3$. show that that $ab = ba$.
106322
Let $A$ and $B$ be square matrices of the same size such that $AB = BA$ and $A$ is nilpotent. Show that $AB$ is nilpotent.
81296
If A and B are ideals of a ring, show that A + B = $\{a+b|a \in A, b \in B\}$ is an ideal
9574
How to show that $\det(AB) =\det(A) \det(B)$?
147332
A finite von Neumann regular ring is unital and has $ab = 1$ if $ba = 1$
149519
Prove that a Boolean ring is a Bezout ring .
66968
if $a,b$ are nilpotent elements of a commutative ring $R$, show that $a+b$ is also nilpotent
53113
$I-AB$ be invertible $\Leftrightarrow$ $I-BA$ is invertible