Let $R$ be a ring satisfying $(xy)^3=xy$ for all $x,\, y\in R$. Then $R$ is commutative.
75520
If $a^3 =a$ for all $a$ in a ring $R$, then $R$ is commutative.
74559
$A$ and $B$ are ideals of a ring ${R}$ such that $A\cap B=\{0\}$. Prove that $st=0$ for every $s\in A, t\in B$.
128493
Given a commutative ring $R$ and an epimorphism of free modules $R^m \to R^n$ is then $m \geq n$?
35777
$\mathbb{R}[X,Y]/(Y-X^2,Y+X)\cong_\text{Ring} \mathbb{R}\times \mathbb{R}$
3129
A commutative ring is a field iff the only ideals are $(0)$ and $(1)$
48537
If $\forall x \in R, x^2-x \in Z(G)$, than $R$ is commutative
151803
Prove that $R$ is commutative
10254
Is the Polynomial Factor Theorem true over all commutative rings?
73689
A question about the equivalence relation on the localization of a ring.
33611
necessary and sufficient condition for trivial kernel of a matrix over a commutative ring
36996
If $x^{2}-x\in Z(R)$ for all $x\in R$, then $R$ is commutative.
6896
If $A/I \cong A/J$ as rings and $I\subseteq J,$ then $I=J.$
121465
$R[x]$ has a subring isomorphic to $R$.
137011
showing $\psi: R\to \mathbb C$ is ring isomorphism.
43020
If $I+J=R$, where $R$ is a commutative rng, prove that $IJ=I\cap J$.
69649
If an ideal contains the multiplicative identity, then it is the whole ring
41629
Structure of ideals in the product of two rings