Show that a ring $R$ is a division ring if and only if, for each nonzero $a\in R$, there is a unique element $b\in R$ such that $aba = a$.
8787
In a ring $(A,+, \cdot)$ if $aba = a$ then $bab = b$ and all non zero elements in $A$ are invertible.
135781
Division ring if and only if it has no proper left ideals.
1425
The Chinese Remainder Theorem for Rings.
16023
Are associates unit multiples in a commutative ring with $1$?
57031
Ring of fractions of $R$ is integral over $R$
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?
79535
Prove that $R$ is a commutative ring if $x^3=x$
142863
Show that every ideal of the matrix ring $M_n(R)$ is of the form $M_n(I)$ where $I$ is an ideal of $R$
22544
Let R be a commutative ring, and let P be a prime ideal of R. Suppose that P has no nontrivial zero divisors in it. Show that R is an integral domain.
150986
For a Noetherian ring $R$, we have $\text{Ass}_R(S^{-1}M)=\text{Ass}_R(M)\cap \{P:P\cap S=\emptyset\}$
68807
How to show that the ring $S/A$ has no zero divisors? (Hungerford, Algebra, Problem 12, Chapter III, Section 2)
41629
Structure of ideals in the product of two rings
79957
If $R$ is an infinite ring, then $R$ has either infinitely many zero divisors, or no zero divisors
127418
If ring $B$ is integral over $A$, then an element of $A$ which is a unit in $B$ is also a unit in $A$.
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.
74559
$A$ and $B$ are ideals of a ring ${R}$ such that $A\cap B=\{0\}$. Prove that $st=0$ for every $s\in A, t\in B$.
4397
Finite rings without zero divisors are division rings.