R is a commutative ring with $1\ne 0$ and $R^m \cong R^n$, then $m=n$
49166
$R^n \cong R^m$ iff $n=m$
118781
product of ideals in matrix ring
133238
Why does $\mathrm{Tor}_0^R(M,N)\cong M\otimes_R N$?
155445
What kind of ring $R$ makes $\mathcal O(\mathrm{Proj}\,R[x_0,\cdots,x_n])=R$?
106584
If $A^{2n}=I$, then $A^n=\pm I$ where $A\in M_n(\mathbb{R})$
26661
Ring homomorphisms from $\mathbb{Z}_n$ to $\mathbb{Z}_m$
151803
Prove that $R$ is commutative
122958
$R/I$ is commutative iff $rs-sr \in I$
63338
Noncommutative failure of IBN: $\,R^n\cong R^m \Rightarrow n - m,$ ring $R$ with maximal ideal $I$ such that $R/I$ is not a division ring?
88307
Commutative property of ring addition
21840
Zero divisors and units for the ring $\mathbb{Z}/n\mathbb{Z}$ with integer $n$
28136
$A^m\hookrightarrow A^n$ implies $m\leq n$ for a ring $A\neq 0$
73689
A question about the equivalence relation on the localization of a ring.
73235
Rings $R/I \cong h(R)/h(I)\,$ for injective ring hom $h$
119067
Basic Ring Theory Question involving the unit element
150986
For a Noetherian ring $R$, we have $\text{Ass}_R(S^{-1}M)=\text{Ass}_R(M)\cap \{P:P\cap S=\emptyset\}$
70320
Let $R$ be a commutative Noetherian ring (with unity), and let $I$ be an ideal of $R$ such that $R/I \cong R$. Then is $I=(0)$?