Show that if $M$ is projective then $\operatorname{Hom}_R(M,.)$ is an exact functor.
121246
Projective modules and $\hom(P,\cdot)$
148111
Every $R$-module is projective if and only if every $R$-module is injective
40705
If $M$ is an $R$-module and $I\subseteq\mathrm{Ann}(M)$ an ideal, then $M$ has a structure of $R/I$-module
48291
Proof that $\mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $
126644
Simple module and homomorphisms
73689
A question about the equivalence relation on the localization of a ring.
104747
Does localisation commute with Hom for finitely-generated modules?
132653
How to show that $\mathfrak{sl}_n(\mathbb{R})$ and $\mathfrak{sl}_n(\mathbb{C})$ are simple?
41895
Show $\text{Hom}_{Ab}(A,\mathbb{Q}) = \{0\}$ for finite $A$
73825
Every $R$-module is a quotient of a free module
83050
Show that $P[x]+\langle x\rangle $ is a prime ideal of $R[x]$.
133238
Why does $\mathrm{Tor}_0^R(M,N)\cong M\otimes_R N$?
39728
integral cohomology ring of real projective space
42552
Localisation is isomorphic to a quotient of polynomial ring
7870
For $M\otimes_R N$, why is it so imperative that $M$ be a right $R$-module and $N$ a left $R$-module?
151705
$SL_n(\mathbb R)$ is a subgroup of kernel of $f\in$ Hom$(GL_n(\mathbb R), A)$, $A$ is an abelian group.
122189
Show that $M[x]$ is a Noetherian $A[x]$-module.