Show that a ring $R$ with $|R|>1$ is a division ring if it satisfies the following condition
8787
In a ring $(A,+, \cdot)$ if $aba = a$ then $bab = b$ and all non zero elements in $A$ are invertible.
79650
Division Rings and Division Algebras
118781
product of ideals in matrix ring
50908
Showing a Ring and an Ideal are equal
138826
To show that either $R$ is a field or $R$ is a finite ring with prime number of elements and $ab = 0$ for all $a,b \in R$.
150986
For a Noetherian ring $R$, we have $\text{Ass}_R(S^{-1}M)=\text{Ass}_R(M)\cap \{P:P\cap S=\emptyset\}$
52966
Rings, ideals and units-proof
117590
Let I and J be ideals of a ring R. Show by example that the set of products {xy | x ∈ I, y ∈ J} need not be an ideal
14873
$1+a$ and $1-a$ in a ring are invertible if $a$ is nilpotent
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.
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 $R$
85913
R is commutative ring with identity & define $\circ$ on $R$ by for any $a,b \in R$ $a \circ b=a+b-ab$ Prove the following
115920
Showing $R$ is a local ring if and only if all elements of $R$ that are not units form an ideal
73689
A question about the equivalence relation on the localization of a ring.
98818
Quotient ring of a localization of a ring
110984
Ideals and prime ideals in a commutative ring.
154613
Showing a set of fractions is a ring