Let $R$ be a ring except that addition and multiplication in $R$ are not assumed to be commutative. Show that addition must be commutative.
70610
Why is ring addition commutative?
76749
Example of a finite non-commutative ring without a unity
62635
Prove that the additive group $ℚ$ is not isomorphic with the multiplicative group $ℚ^*$.
18971
Show that a ring is commutative if it has the property that ab = ca implies b = c when $a\neq 0$
114640
Showing that an ideal is not a free $R-$module.
22544
Let R be a commutative ring, and let P be a prime ideal of R. Suppose that P has no nontrivial zero divisors in it. Show that R is an integral domain.
75558
Counter-example for prime ideals in a non commutative ring.
127030
Simple example of non-arithmetic ring (non-distributive ideal lattice)
116916
What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring?
79535
Prove that $R$ is a commutative ring if $x^3=x$
73248
Prove that $\mathbb{R^*}$, the set of all real numbers except $0$, is not a cyclic group
151803
Prove that $R$ is commutative
92962
Rings with $a^5=a$ are commutative
115920
Showing $R$ is a local ring if and only if all elements of $R$ that are not units form an ideal
28327
(Simple) Examples on Non-Commutative Rings
11938
What happens if we remove the requirement that $\langle R, + \rangle$ is abelian from the definition of a ring?
69353
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?