Does $ak \equiv bk \textrm{ mod }m \implies a \equiv b \textrm{ mod } m$?
14693
$d = \gcd(c,m),\ ac≡bc\pmod{\! m}\Rightarrow a≡b\pmod {\!m/d}\ $ [Congruence Cancellation & Division Rule]
14259
If M+N and M$\cap$N are finitely generated modules, so are M and N.
50114
A question about direct sums and products of modules
92211
When are $\mathbb Z_m$ and $\mathbb Z_n$ homomorphic?
138058
Prove that $\mathbb Z_{m}\times\mathbb Z_{n} \cong \mathbb Z_{mn}$ implies $\gcd(m,n)=1$.
8374
How to prove that $R/I \otimes_R M \cong M / IM$
11989
$ac\equiv bc \pmod{\!m}\!\!\iff\!\! a\equiv b \pmod{\!\!\frac{m}{(c,m)}}$ [Euclid's Lemma, congruence form]
48291
Proof that $\mathrm{ker}(f^*) = \mathrm{im}(f)^{\perp} $
82921
Uniqueness of $k$th root mod $m$ if $(k, \phi(m)) = 1$.
75614
Proving that $n|m\implies f_n|f_m$
92518
mod Distributive Law, factoring $\!\!\bmod\!\!:$ $\ ab\bmod ac = a(b\bmod c)$
100303
A necessary and sufficient condition for $A\cap(B+C)=(A\cap B)+C$ to hold for submodules
145722
Prove there is a bijection between the set of $R[x]$-module structures on $M$ and $\mathrm{End}_{R-\mathsf{Mod}}(M)$.
138088
Diffie-Hellman key exchange: Why is $(g^{k_1} \bmod n )^{k_2} \bmod n \equiv (g^{k_2} \mod n)^{k_1} \bmod n$
55900
Is there a ring homomorphism $M_2(\mathbb Z)\to \mathbb Z$?
21250
Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
46147
Proving $(\forall a\in\mathbb{Z^+})(m\in\mathbb{Z^+}\to a^m\equiv a^{m-\phi(m)}\pmod{m})$