query: Show that the ring is commutative
positive_passages.docid: 75520
positive_passages.text: If $a^3 =a$ for all $a$ in a ring $R$, then $R$ is commutative.
negative_passages.docid: 27766
negative_passages.text: Comaximal ideals in a commutative ring
negative_passages.docid: 135699
negative_passages.text: Questions about a commutative ring with exactly three ideals
negative_passages.docid: 138337
negative_passages.text: On a ring $R$ where $x^2 = x$, we must have that $2x = 0$ and $R$ is commutative
negative_passages.docid: 67470
negative_passages.text: If $S$ is a ring with the property that $s=s^2$ for each $s \in S$, then is $S$ commutative?
negative_passages.docid: 53770
negative_passages.text: Let $R$ be a finite commutative ring. Show that an ideal is maximal if and only if it is prime.
negative_passages.docid: 122189
negative_passages.text: Show that $M[x]$ is a Noetherian $A[x]$-module.
negative_passages.docid: 19838
negative_passages.text: Prove that in general the set of the right zero divisors is different from the set of left zero divisors in a ring R
negative_passages.docid: 108930
negative_passages.text: Show that an operation being commutative is a structural property
negative_passages.docid: 79535
negative_passages.text: Prove that $R$ is a commutative ring if $x^3=x$
negative_passages.docid: 142612
negative_passages.text: Matrix Ring of a Semisimple Ring
negative_passages.docid: 138826
negative_passages.text: To show that either $R$ is a field or $R$ is a finite ring with prime number of elements and $ab = 0$ for all $a,b \in R$.
negative_passages.docid: 85913
negative_passages.text: 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
negative_passages.docid: 1686
negative_passages.text: Show that quotient rings are not isomorphic
negative_passages.docid: 88307
negative_passages.text: Commutative property of ring addition
negative_passages.docid: 69353
negative_passages.text: Show that a ring with only trivial right ideals is either a division ring or $|R|=p$ and $R^2=\{0\}$. Why would $R$ be finite?
negative_passages.docid: 92517
negative_passages.text: Easy way to show that $\mathbb{Z}[\sqrt[3]{2}]$ is the ring of integers of $\mathbb{Q}[\sqrt[3]{2}]$