If $A$ is an ideal of $R$, show that $R/A$ is commutative if and only if $rs - sr \in A$ for all $r, s\in R$.
122958
$R/I$ is commutative iff $rs-sr \in I$
92962
Rings with $a^5=a$ are commutative
50908
Showing a Ring and an Ideal are equal
64484
Left ideal $I$ is a direct summand of $R$ iff $I = Rr$, $r^2 = r$
63171
Translate of an ideal by $a$ coincides with the ideal iff  $a \in I$
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?
83009
Showing that $R(x)$ is a proper subset of $R((x))$ if $R$ is a field
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
58640
$R$ is an algebra over an infinite field. If $\exists$ ideals s.t. $J\subseteq \bigcup_{k=1}^nI_k$ then $J\subseteq I_k$ for some $k$
29126
Maximal ideal in the ring of continuous functions from $\mathbb{R} \to \mathbb{R}$
70320
Let $R$ be a commutative Noetherian ring (with unity), and let $I$ be an ideal of $R$ such that $R/I \cong R$. Then is $I=(0)$?
103518
If $J$ is the ideal generated by all idempotents in a prime ideal, then $R/J$ has only trivial idempotents
77198
Showing that the characteristic of a commutative ring R without zero divisors is 0 or prime
111307
If ideal $I$ of domain $R$ is free $R$-module, then $I$ is principal ideal.
3129
A commutative ring is a field iff the only ideals are $(0)$ and $(1)$
79897
if R is a commutative ring in which all the prime ideals are finitely generated then R is Noetherian
150986
For a Noetherian ring $R$, we have $\text{Ass}_R(S^{-1}M)=\text{Ass}_R(M)\cap \{P:P\cap S=\emptyset\}$