query: Let $A$ be a ring, and $Z(A)$ its center. Show that if $x^2-x \in Z(A)$ for every $x \in A$ then $A$ is commutative
positive_passages.docid: 36996
positive_passages.text: If $x^{2}-x\in Z(R)$ for all $x\in R$, then $R$ is commutative.
negative_passages.docid: 116943
negative_passages.text: Proof for maximal ideals in $\mathbb{Z}[x]$
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: 75520
negative_passages.text: If $a^3 =a$ for all $a$ in a ring $R$, then $R$ is commutative.
negative_passages.docid: 14873
negative_passages.text: $1+a$ and $1-a$ in a ring are invertible if $a$ is nilpotent
negative_passages.docid: 77198
negative_passages.text: Showing that the characteristic of a commutative ring R without zero divisors is 0 or prime
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: 43990
negative_passages.text: R is a commutative ring with $1\ne 0$ and $R^m \cong R^n$, then $m=n$
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: 108930
negative_passages.text: Show that an operation being commutative is a structural property
negative_passages.docid: 37393
negative_passages.text: $R$ is commutative, $I$,$J$ are ideals, $I+J=R$, then $IJ=I\cap J$
negative_passages.docid: 33728
negative_passages.text: Basis for Center of Group Ring
negative_passages.docid: 49121
negative_passages.text: Factorizing a polynomial $f$ in $A[x]$ (with $A$ commutative), where $f$ has a zero in its field of fractions
negative_passages.docid: 110984
negative_passages.text: Ideals and prime ideals in a commutative ring.
negative_passages.docid: 114343
negative_passages.text: 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 $m,n \in \mathbb{N}$
negative_passages.docid: 148788
negative_passages.text: Polynomial ring ideal. Show that $(f) \cap R_N = (0) \iff f \notin R_N$
negative_passages.docid: 135699
negative_passages.text: Questions about a commutative ring with exactly three ideals