Where does this proof of $R^n\cong R^m\Rightarrow m=n$ break down for noncommutative rings?
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?
91947
Proof of $ f:I_n \rightarrow I_m \Rightarrow n = m$
102072
Simple $M_n(D)$-module with $D$ a division ring
134294
A (not necessarily unitary) $R$ module $M$ splits into $B\oplus C$ where $B$ is unitary and $RC=\{0\}$.
109470
Proof of whether a specific element is in a commutative, invertible monoid.
87946
Proving that the unities of a ring form a group under multiplication
16602
Proving $M_n(I)\lhd M_n(R)$ and proofs about ideal rings.
71822
Commutator algebra in exponents
97608
Showing a Ring of endomorphisms is isomorphic to a Ring
52642
Show that $\text{C}/c \cong (c/(\text{C}^{\text{op}}))^{\text{op}}$
14581
How do we prove $n^n \mid m^m \Rightarrow n \mid m$?
151803
Prove that $R$ is commutative
114343
If $I$ and $J$ are ideals in a ring $R$ with $1$ such that $I+J = R$, show that $I^m$ and $J^n$ are co-maximal for all $m,n \in \mathbb{N}$
39453
How do I prove that $S^n$ is homeomorphic to $S^m \Rightarrow m=n$?
88307
Commutative property of ring addition
98352
If $k|n$, then $D_{2n}/\langle{r^k}\rangle\cong D_{2k}$
147564
Why does $(A/I)\otimes_R (B/J)\cong(A\otimes_R B)/(I\otimes_R 1+1\otimes_R J)$?