Let R be a ring in which $x^2=x$ for all x ε R. (a) Prove that $x + x = 0$ for for all x ε R. (b) Prove that R is commutative.
65224
Prove that if $a^2 = a$ for all elements of the ring $R$, then $R$ has characteristic 2. Is the converse of the statement true?
98630
For all $x,y\in\mathbb{R}$ define that $x\equiv y$ if $x^{2}=y^{2} $; prove $\equiv$ is an equivalence relation.
92962
Rings with $a^5=a$ are commutative
30327
$\ker \phi = (x_1-a_1, ..., x_n-a_n)$ for a ring homomorphism $\phi: R[x_1, ..., x_n] \to R$
59687
Suppose $M$ is a finitely generated module over $R$ where $R$ is a Noetherian ring. Prove that $M$ is a noetherian module.
10254
Is the Polynomial Factor Theorem true over all commutative rings?
118873
Prove that $f(x)=0$ for all $x\in\mathbb{R}$.
106894
If a subring of a ring R has identity, does R also have the identity?
121750
Let $R$ be a commutative, unital ring and let $I$ be an ideal of $R.$ Prove that if $I$ is maximal then $R/I$ is a field.
117590
Let I and J be ideals of a ring R. Show by example that the set of products {xy | x ∈ I, y ∈ J} need not be an ideal
146231
Let X be a set with an associative binary operation ◦ with identity e. Suppose every element of X satisﬁes x ◦ x = e. Prove that ◦ is commutative.
155445
What kind of ring $R$ makes $\mathcal O(\mathrm{Proj}\,R[x_0,\cdots,x_n])=R$?
87078
Prove the following set is an ideal of the ring $Z[x]$
77198
Showing that the characteristic of a commutative ring R without zero divisors is 0 or prime
78626
Let $R$ be a commutative ring. If $R[X]$ is a principal ideal domain, then $R$ is a field.
101543
Let $A$ be a group, where $a^2=1$, a belongs to $A$. Prove that this group is commutative.
111493
Division algorithm for polynomials in R[x], where R is a commutative ring with unity.