$R^n \cong R^m$ if and only if $n=m$ without assuming the result for vector spaces
28136
$A^m\hookrightarrow A^n$ implies $m\leq n$ for a ring $A\neq 0$
8291
If $n$ vectors are linearly independent, is their span $\mathbb{R}^n$?
65371
Does $2^X \cong 2^Y$ imply $X \cong Y$ without assuming the axiom of choice?
39453
How do I prove that $S^n$ is homeomorphic to $S^m \Rightarrow m=n$?
51145
Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets?
24561
$\mathbb S_n$ as semidirect product
53266
Show that $R/I\otimes_R R/J\cong R/(I+J)$
118898
Is there a simple way of proving that $\text{GL}_n(R) \not\cong \text{GL}_m(R)$?
128711
$(\Bbb Z\oplus \Bbb Z)[X] \not\cong \Bbb Z[X]\oplus \Bbb Z[X]\,$?
141258
Why does $M \mathbin{\otimes_R} N \cong M_\mathfrak{p} \mathbin{\otimes_{R_\mathfrak{p}}} N$?
136756
Computational check whether $G\cong \tilde G$
5223
$\mathbb{Z}_m \times \mathbb Z_n$ is cyclic if and only if $\gcd(m,n)=1$
4648
Proving that $\mathbb{Z}_m\oplus \mathbb{Z}_n \cong \mathbb{Z}_d\oplus \mathbb{Z}_l $ as groups, where $l=\mathrm{lcm}(m,n)$ and $d=\gcd(m,n)$
35948
How to prove $D^n/S^{n-1}\cong S^n$?
46355
If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
134365
When is this true: If $m^2$ is a multiple of N then m is a multiple of N
141799
Is $2^{|\mathbb{N}|} = |\mathbb{R}|$?