Prove that for any n, $[k]_n$ is invertible in the ring $\mathbb{Z}_n$ if and only if $n$ and $k$ are relatively prime.
31917
If $q$ is coprime to $a$ then $a\mid nq-1,\,$ so $q$ is invertible mod $a$
64742
Prove a residue matrix $A$ (with coefficients in $\mathbb Z_n)$ has an inverse if and only if $\gcd(\det A,n) = 1$
8787
In a ring $(A,+, \cdot)$ if $aba = a$ then $bab = b$ and all non zero elements in $A$ are invertible.
119648
For which $n$ and $k$ are there nontrivial homomorphisms from $S_n$ to $\mathbb{Z}/k\mathbb{Z}$?
5223
$\mathbb{Z}_m \times \mathbb Z_n$ is cyclic if and only if $\gcd(m,n)=1$
57669
A proof that $\mathbb{Z}_{p}$ is an integral domain if and only if $p$ is prime.
119297
Finding invertible polynomials in polynomial ring $\mathbb{Z}_{n}[x]$
14873
$1+a$ and $1-a$ in a ring are invertible if $a$ is nilpotent
143543
Prove that $(n - 1)^2 \mid n^k -1$ if and only if $(n - 1) \mid k$
118073
Is the ring $m\mathbb{Z}$ isomorphic to the ring $n\mathbb{Z}$?
112701
Prove that if matrix $A$ is nilpotent, then $I+A$ is invertible.
111114
Prove that an ideal in $k[x,y]$ is a prime ideal
45523
Can the product of two non invertible elements in a ring be invertible?
25229
Can any set of $n$ relatively prime elements be extended to an invertible matrix?
138058
Prove that $\mathbb Z_{m}\times\mathbb Z_{n} \cong \mathbb Z_{mn}$ implies $\gcd(m,n)=1$.
146795
Proof If $AB-I$ Invertible then $BA-I$ invertible.
118452
how to prove $\mathbb Z_n$ has divisors of zero if and only if $n$ is not prime