If $a$ is an element of some ring and $\{a^n; n=0,1,\dots\}$ is finite, then $a$ is invertible or a zero divisor
14343
Every nonzero element in a finite ring is either a unit or a zero divisor
45523
Can the product of two non invertible elements in a ring be invertible?
8494
Proof Check: Number of elements of $\mathbb{F}_{p^{n}}$ of the form $a^{p}-a$ for some $a \in \mathbb{F}_{p^{n}}$.
107378
If a ring element is right-invertible, but not left-invertible, then it has infinitely many right-inverses.
125604
Proof that $\mathbb{Z}$ has no zero divisors
4397
Finite rings without zero divisors are division rings.
128043
$A^n=0$ for $A$ nilpotent without the use of minimal polynomial
146304
prove $(n)$ prime ideal of $\mathbb{Z}$ iff $n$ is prime or zero
77198
Showing that the characteristic of a commutative ring R without zero divisors is 0 or prime
114254
Zero divisors in $A[x_1,x_2,\dots,x_r]$
151928
How to prove that $N\setminus A$ is finite?
32108
$ a\in \Bbb Z_n$ is invertible $\,\Rightarrow\gcd (a,n) =1$
24203
Finding Zero Divisors and Units of Rings $\mathbb{Z}_n \times \mathbb{Z}_m$
3129
A commutative ring is a field iff the only ideals are $(0)$ and $(1)$
22666
Proving that if $A^n = 0$, then $I - A$ is invertible and $(I - A)^{-1} = I + A + \cdots + A^{n-1}$
50908
Showing a Ring and an Ideal are equal
92822
Finite number of elements generating the unit ideal of a commutative ring