query: Suppose H is a subring of R such that for every $r∈H$ we have $r^2=e$. Prove that $H$ is commutative.
positive_passages.docid: 20679
positive_passages.text: Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
negative_passages.docid: 156878
negative_passages.text: If every commutator is idempotent, then the ring is commutative
negative_passages.docid: 35800
negative_passages.text: If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
negative_passages.docid: 48537
negative_passages.text: If $\forall x \in R, x^2-x \in Z(G)$, than $R$ is commutative
negative_passages.docid: 100987
negative_passages.text: Let f be an r-cycle in $S_n$. Given any $h \in S_n$, show that $hfh^{-1} = (h(x_1), h(x_2), \dots, h(x_r))$
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: 124672
negative_passages.text: Let $f:R\rightarrow S$ be a ring homomorphism. Prove or disprove: if $f$ is onto and $R$ is an integral domain, then $S$ is an integral domain
negative_passages.docid: 80212
negative_passages.text: Suppose that $X$ is a topological space and $x_0\in X$. Prove that $\widetilde{H_n}(X)=H_n(X,x_0)$ for all $n\geq 0$.
negative_passages.docid: 145397
negative_passages.text: How to prove that for every $i$, $a_i = i$
negative_passages.docid: 110984
negative_passages.text: Ideals and prime ideals in a commutative ring.
negative_passages.docid: 105471
negative_passages.text: How to prove that if each element of group is inverse to itself then group commutative?
negative_passages.docid: 71822
negative_passages.text: Commutator algebra in exponents
negative_passages.docid: 92962
negative_passages.text: Rings with $a^5=a$ are commutative
negative_passages.docid: 101543
negative_passages.text: Let $A$ be a group, where $a^2=1$, a belongs to $A$. Prove that this group is commutative.
negative_passages.docid: 155579
negative_passages.text: Show that $End_A(A)$ = {$r_a$ | $a ∈ A$}
negative_passages.docid: 149747
negative_passages.text: Prove that if I is a prime ideal of R then I[x] is a prime ideal of R[x]
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 $a\neq 0$