query: Let R be a ring. Prove that if $x^2=x$ for each $x \in R$, then R is a commutative ring.
positive_passages.docid: 156878
positive_passages.text: If every commutator is idempotent, then the ring is commutative
negative_passages.docid: 35777
negative_passages.text: $\mathbb{R}[X,Y]/(Y-X^2,Y+X)\cong_\text{Ring} \mathbb{R}\times \mathbb{R}$
negative_passages.docid: 69728
negative_passages.text: If $f$ divides $g$ in $S[x]$, show that $f$ divides $g$ in $R[x]$ for $R$ a sub-ring of $S$.
negative_passages.docid: 18971
negative_passages.text: Show that a ring is commutative if it has the property that ab = ca implies b = c when $a\neq 0$
negative_passages.docid: 89702
negative_passages.text: Let $R$ be a ring such that for all $a,b$ in $R$, $(a^2-a)b=b(a^2-a)$. Then $R$ is commutative
negative_passages.docid: 149519
negative_passages.text: Prove that a Boolean ring is a Bezout ring .
negative_passages.docid: 136985
negative_passages.text: Every finite commutative ring $R$ must be of the form
negative_passages.docid: 155445
negative_passages.text: What kind of ring $R$ makes $\mathcal O(\mathrm{Proj}\,R[x_0,\cdots,x_n])=R$?
negative_passages.docid: 70320
negative_passages.text: 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 $I=(0)$?
negative_passages.docid: 27766
negative_passages.text: Comaximal ideals in a commutative ring
negative_passages.docid: 87078
negative_passages.text: Prove the following set is an ideal of the ring $Z[x]$
negative_passages.docid: 102316
negative_passages.text: why $(r+I)(s+I) = rs + I$ in the quotient ring $R ?$
negative_passages.docid: 52966
negative_passages.text: Rings, ideals and units-proof
negative_passages.docid: 19112
negative_passages.text: Ring $R= \mathbb Z_3[x]/\langle x^6-1\rangle$
negative_passages.docid: 158311
negative_passages.text: What is the dimension of $R[[X]]$ where $R$ is a Noetherian ring?
negative_passages.docid: 88307
negative_passages.text: Commutative property of ring addition
negative_passages.docid: 83100
negative_passages.text: Prove that the center of a ring is a subring.