$R$ is an associative ring with no zero divisors that has an element $a\neq 0$, such that $a*a=a$. Prove $a$ is the identity element of $R$
32741
Idempotents in a ring without unity (rng) and no zero divisors.
115920
Showing $R$ is a local ring if and only if all elements of $R$ that are not units form an ideal
43990
R is a commutative ring with $1\ne 0$ and $R^m \cong R^n$, then $m=n$
106894
If a subring of a ring R has identity, does R also have the identity?
134989
Prove that if the identity is written as the product of $r$ transpositions, then $r$ is an even number
86977
If $\cup \mathcal{F}=A$ then $A \in \mathcal{F}$. Prove that $A$ has exactly one element.
66968
if $a,b$ are nilpotent elements of a commutative ring $R$, show that $a+b$ is also nilpotent
114254
Zero divisors in $A[x_1,x_2,\dots,x_r]$
16023
Are associates unit multiples in a commutative ring with $1$?
61274
If $AA^T$ is the zero matrix, then $A$ is the zero matrix
74924
identity element of $(D_n ,\gcd)$
155445
What kind of ring $R$ makes $\mathcal O(\mathrm{Proj}\,R[x_0,\cdots,x_n])=R$?
74441
Zero divisors in matrix rings
6646
A (non-artificial) example of a ring without maximal ideals
33052
How to prove $\bar{m}$ is a zero divisor in $\mathbb{Z}_n$ if and only if $m,n$ are not coprime
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.
102316
why $(r+I)(s+I) = rs + I$ in the quotient ring $R ?$