Decompose finitely generated modules and use Krull-Schmidt theorem
133915
In which case $M_1 \times N \cong M_2 \times N \Rightarrow M_1 \cong M_2$ is true?
69398
Rank-nullity theorem for free $\mathbb Z$-modules
34065
Submodule of free module over a p.i.d. is free even when the module is not finitely generated?
127044
Prove that a finitely-generated free module has a finite basis
117211
Infinite linear independent family in a finitely generated $A$-module
116955
Showing the existence of a certain kind of extension of a finitely generated algebra
1425
The Chinese Remainder Theorem for Rings.
130186
Zero module of differentials implies finite extension?
47630
Endomorphisms of modules satisfying chain conditions and counterexamples.
4790
Generators of a finitely generated free module over a commutative ring
81550
Linking two theorems: on algebraic closure and on minimal splitting field
145722
Prove there is a bijection between the set of $R[x]$-module structures on $M$ and $\mathrm{End}_{R-\mathsf{Mod}}(M)$.
148828
Submodule of a free module
73825
Every $R$-module is a quotient of a free module
70658
Isomorphism between quotient modules
123130
Showing an isomorphism between exterior products of a free module
37570
$M + N$ and $M \cap N$ are finitely generated $A$-modules implies $M$ and $N$ finitely generated using exact sequences