If R is a ring with unity , suppose $x \neq 0\in R, \exists $a unique $y \in R$ such that xyx = x, then show that xy = yx = 1
8787
In a ring $(A,+, \cdot)$ if $aba = a$ then $bab = b$ and all non zero elements in $A$ are invertible.
76110
Proving that the zero vector is unique
111493
Division algorithm for polynomials in R[x], where R is a commutative ring with unity.
44100
Prove that identity element is unique
158426
Proving lemma about centrality of idempotent elements in a Ring with no nilpotent elements.
31933
$x^2 = 1$ in the ring $\mathbb{Z}/n \mathbb{Z}$
155604
Show that $k[x,y]/(xy-1)$ is not isomorphic to a polynomial ring in one variable.
32772
Proving units in a ring
87013
If $\gcd(a,b)=1$, then there exists integers $x$ and $y$ such that $xa + yb = 1$
43990
R is a commutative ring with $1\ne 0$ and $R^m \cong R^n$, then $m=n$
1990
Results for elements in Dihedral Groups: $x \notin \langle r\rangle  \Rightarrow rx = xr^{-1}$
18971
Show that a ring is commutative if it has the property that ab = ca implies b = c when $a\neq 0$
100845
How to prove that $G(x)=ax+b$ is a one-to-one correspondence where $a\neq0$ and $a,b\in\mathbb{R}$.
18108
Show if $(a,p)=1$ there is a unique inverse of $a$ modulo $p$
34183
An equivalent condition with $\{0\}$ being the only nilpotent ideal
154327
$A^2X = X$ show that $AX = X$, if $A_{ij} > 0$ and $x_{i} > 0$
30327
$\ker \phi = (x_1-a_1, ..., x_n-a_n)$ for a ring homomorphism $\phi: R[x_1, ..., x_n] \to R$