query: Prove that $R$ 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: 23095
negative_passages.text: Prove that T is a linear transformation
negative_passages.docid: 66968
negative_passages.text: if $a,b$ are nilpotent elements of a commutative ring $R$, show that $a+b$ is also nilpotent
negative_passages.docid: 102316
negative_passages.text: why $(r+I)(s+I) = rs + I$ in the quotient ring $R ?$
negative_passages.docid: 72251
negative_passages.text: $g\circ f$ is Surjective and $g$ is Injective then Prove that $f$ is surjective
negative_passages.docid: 34768
negative_passages.text: Proving $(p \oplus q) \oplus r=p \oplus (q \oplus r)$
negative_passages.docid: 89125
negative_passages.text: Prove that $r \in R$ such that $0 \lt r \lt 1$ then $\frac{1}{r (1 - r)} \ge 4$
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: 122958
negative_passages.text: $R/I$ is commutative iff $rs-sr \in I$
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: 69875
negative_passages.text: Prove the commutativity property of addition of natural numbers by induction
negative_passages.docid: 8374
negative_passages.text: How to prove that $R/I \otimes_R M \cong M / IM$
negative_passages.docid: 137011
negative_passages.text: showing $\psi: R\to \mathbb C$ is ring isomorphism.
negative_passages.docid: 52966
negative_passages.text: Rings, ideals and units-proof
negative_passages.docid: 53266
negative_passages.text: Show that $R/I\otimes_R R/J\cong R/(I+J)$
negative_passages.docid: 13983
negative_passages.text: Show that If $R[X]$ is Euclidean domain then $R$ is a field
negative_passages.docid: 35777
negative_passages.text: $\mathbb{R}[X,Y]/(Y-X^2,Y+X)\cong_\text{Ring} \mathbb{R}\times \mathbb{R}$