Prove that if $R$ is a non-commutative ring with $1$ and if $a,b \in R$ and if $ab =1 $ but $ba \neq 1$ then $R$ is infinite
107378
If a ring element is right-invertible, but not left-invertible, then it has infinitely many right-inverses.
117590
Let I and J be ideals of a ring R. Show by example that the set of products {xy | x ∈ I, y ∈ J} need not be an ideal
66968
if $a,b$ are nilpotent elements of a commutative ring $R$, show that $a+b$ is also nilpotent
130073
$N(ab) \geq N(a)$ is not necessary for the ring to be euclidean
131671
Let R be a ring such that $a^2 = a$ , $\forall a$ $\in R$ . Prove that R is commutative.
135699
Questions about a commutative ring with exactly three ideals
6646
A (non-artificial) example of a ring without maximal ideals
157863
A finite pseudo-ring such that $ab^2=b$ for some $(a,b)\in A$
149519
Prove that a Boolean ring is a Bezout ring .
74559
$A$ and $B$ are ideals of a ring ${R}$ such that $A\cap B=\{0\}$. Prove that $st=0$ for every $s\in A, t\in B$.
2709
How to show that every Boolean ring is commutative?
134991
Commutative non Noetherian rings in which all maximal ideals are finitely generated
58640
$R$ is an algebra over an infinite field. If $\exists$ ideals s.t. $J\subseteq \bigcup_{k=1}^nI_k$ then $J\subseteq I_k$ for some $k$
110944
Proving that $AB-BA=cI$ for nontrivial $c \in \mathbb{C}$
91962
In a ring $a*a=0$ then $a+a=0$
79535
Prove that $R$ is a commutative ring if $x^3=x$
103349
A question about a proof of Noetherian modules and exact sequences