Sufficient condition for a ring to be a product of two rings
16255
If $\mathop{\mathrm{Spec}}A$ is not connected then there is a nontrivial idempotent
28362
Product of two cyclic groups is cyclic iff their orders are co-prime
91777
Product of Non-Disjoint Cycles
51428
Do these two observations suffice to show that a finite boolean ring must be of the form $\mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$?
114688
Show that a division ring is simple.
33611
necessary and sufficient condition for trivial kernel of a matrix over a commutative ring
138337
On a ring $R$ where $x^2 = x$, we must have that $2x = 0$ and $R$ 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?
130814
Quotient of semi-simple rings
100303
A necessary and sufficient condition for $A\cap(B+C)=(A\cap B)+C$ to hold for submodules
18971
Show that a ring is commutative if it has the property that ab = ca implies b = c when $a\neq 0$
10908
Associated elements in a ring
72507
Does $ab=1$ imply $ba=1$ in a ring?
15686
Product of a Lie ideal with a subalgebra
81296
If A and B are ideals of a ring, show that A + B = $\{a+b|a \in A, b \in B\}$ is an ideal
63192
A chain ring with Krull dimension greater than one
114343
If $I$ and $J$ are ideals in a ring $R$ with $1$ such that $I+J = R$, show that $I^m$ and $J^n$ are co-maximal for all $m,n \in \mathbb{N}$