$R$ finite commutative ring with identity. Prove that $a \in R$ is either a zero divisor or a unit.
14343
Every nonzero element in a finite ring is either a unit or a zero divisor
34203
if $S$ is a ring (possibly without identity) with no proper left ideals, then either $S^2=0$ or $S$ is a division ring.
29452
Divisor in $\mathbb{C}[X]$ $\implies$ divisor in $\mathbb{R}[X]$?
89702
Let $R$ be a ring such that for all $a,b$ in $R$, $(a^2-a)b=b(a^2-a)$. Then $R$ is commutative
17926
Zero divisor in $R[x]$
148550
Units in quotient ring of $\mathbb Z[X]$
21840
Zero divisors and units for the ring $\mathbb{Z}/n\mathbb{Z}$ with integer $n$
113512
Characteristic of nonzero commutative rings with unity
116000
Prove that prime ideals of a finite ring are maximal
4397
Finite rings without zero divisors are division rings.
79535
Prove that $R$ is a commutative ring if $x^3=x$
23641
Quotient ring of field is zero or itself.
36996
If $x^{2}-x\in Z(R)$ for all $x\in R$, then $R$ is commutative.
65224
Prove that if $a^2 = a$ for all elements of the ring $R$, then $R$ has characteristic 2. Is the converse of the statement true?
13162
Prove $\alpha \in R[[x]]$ is a unit iff $a_0 \in R$ is a unit
63568
Ring with finitely many zerodivisors
154530
Rings whose elements are partitioned between units and zero-divisors.