Proving two free modules are isomorphic iff they have the same rank
49166
$R^n \cong R^m$ iff $n=m$
40996
Proving that free modules are flat (without appealing projective modules)
120035
Pattern in Isomorphisms between $\mathbb Z_m \times \mathbb Z_n$ and $\mathbb Z_{mn}$
36980
Prove the third isomorphism theorem
145462
On the canonical isomorphism between $V$ and $V^{**}$
52992
Proving that P/PJ is a projective right module over R/J
93118
How to show that if two integral domains are isomorphic, then their corresponding field of quotients are isomorphic?
79591
Exact Sequences of Modules and Rank
115649
Proving two sets $A, B$ are order isomorphic
26570
Prove $\operatorname{rank}(AB) = \operatorname{rank}(B)$
127044
Prove that a finitely-generated free module has a finite basis
120968
Isomorphism of stabilizer over $S_n$ to $S_{n-1}$
116594
Finite fields are isomorphic
17978
Isomorphism classes of $\mathbb{Z}[i]$ modules.
141911
Prove that these two fields are isomorphic.
117211
Infinite linear independent family in a finitely generated $A$-module
52354
Given a idempotent endomorphism, show that $M = \ker(f) \oplus \ker (f-\text{id}_M)$