How to prove $M \otimes_R k\cong M/IM$?
8374
How to prove that $R/I \otimes_R M \cong M / IM$
135808
Show that there is a positive integer $n$ such that $G \cong \ker(\phi^{n}) \times \phi^{n}(G)$
133238
Why does $\mathrm{Tor}_0^R(M,N)\cong M\otimes_R N$?
141258
Why does $M \mathbin{\otimes_R} N \cong M_\mathfrak{p} \mathbin{\otimes_{R_\mathfrak{p}}} N$?
156910
Necessary and sufficient condition for $R/IJ \cong R/I \times R/J$
94469
How to prove that $|HK| = \dfrac{|H| \; |K|}{|H \cap K|}$?
130367
Proof that $K\otimes_F L$ is not noetherian
16172
My proof of $I \otimes N \cong IN$ is clearly wrong, but where have I gone wrong?
36747
$\text{Tor}_0^R(M,N)\simeq M\otimes_R N$
132653
How to show that $\mathfrak{sl}_n(\mathbb{R})$ and $\mathfrak{sl}_n(\mathbb{C})$ are simple?
8417
Natural $(R/I)$-module structure for an $R$-module $M$ annihilated by $I$
65873
A proof for Atiyah-Macdonald Exercise I.21.iii
106401
Prove: $p-mq | f(m)$ where $m$ is any integer
3203
Quotient group $\mathbb Z^n/\ \text{im}(A)$
111746
$(A/\mathfrak{a}) \otimes_A M \simeq M / \mathfrak{a}M$ by tensoring the canonical exact sequence.
120594
$\mathbb Z[i]/ \langle 1+2i \rangle \cong \mathbb Z_5$
99376
Prove that If $m'$ is a common multiple of $s$ and $t$, then $m | m'$. Here $m$ is the LCM of $s$ and $t$.