$M_1 \to M_2 \to M_3 \to 0$ exact iff $0 \to \text{Hom}_A(M_3, N) \to \text{Hom}_A(M_2, N) \to \text{Hom}_A(M_1, N)$ is exact.
25576
Characterization of short exact sequences
38549
$\mathbb{Q}(\sqrt{n}) \cong \mathbb{Q}(\sqrt{m})$ iff $n=m$
91962
In a ring $a*a=0$ then $a+a=0$
111746
$(A/\mathfrak{a}) \otimes_A M \simeq M / \mathfrak{a}M$ by tensoring the canonical exact sequence.
70266
1-1 correspondence between [0,1] and [0,1)
51898
Show that $\operatorname{Hom}_{R} (R,M) \cong M$
43990
R is a commutative ring with $1\ne 0$ and $R^m \cong R^n$, then $m=n$
2342
How does equality $\mathfrak{m}^\mathfrak{m} = 2^\mathfrak{m}$ in $\sf{ZF}$ relate to the axiom of choice?
23071
Showing that $f_0 (x_1, \ldots, x_m) \mathrm tr A =\displaystyle{ \sum_{i=1}^n} f_0(x_1, \ldots, Ax_i,\ldots, x_m)$
72551
Proof that $\rho = \pm1$ implies linear combination
150986
For a Noetherian ring $R$, we have $\text{Ass}_R(S^{-1}M)=\text{Ass}_R(M)\cap \{P:P\cap S=\emptyset\}$
70620
From $n=2$ to $n$-ary Euclid lemma $\ p\mid a_1\cdots a_n\Rightarrow\ p\mid a_1\,$ or $\,\cdots\, p\mid a_n$
134365
When is this true: If $m^2$ is a multiple of N then m is a multiple of N
63171
Translate of an ideal by $a$ coincides with the ideal iff  $a \in I$
29204
$a\equiv\bar a\!\pmod{\!kn}\Rightarrow a\equiv\bar a\!\pmod{\! n};\ $ $(a\bmod kn)\bmod n=a\bmod n.\ $ Congruences persist mod factors of the modulus
135986
Show if $a \equiv b \pmod{m}$ then $ac \equiv bc\pmod{mc}$
38378
$M= \{ A \in Mat_{2 \times 2}{\mathbb{R}}| \det(A)=1 \}$ is homeomorphic to $S^{1} \times \mathbb{R}^{2}$