A bijective ring homomorphism is called an isomorphism.

If there is an isomorphism from $R$ to $S$, then we say that rings $R$ and $S$ are isomorphic (as rings).

Proof.

Suppose that the rings are isomorphic. Then we have a ring isomorphism
\[f:2\Z \to 3\Z.\]
Let us put $f(2)=3a$ for some integer $a$. Then we compute $f(4)$ in two ways.
First we have
\[f(4)=f(2+2)=f(2)+f(2)=3a+3a=6a.\]

Next we have
\[f(4)=f(2\cdot 2)=f(2)\cdot f(2)=3a\cdot 3a=9a^2.\]
These are equal and hence we have
\[6a=9a^2.\]
The only integer solution is $a=0$.

But then we have $f(0)=0=f(2)$, which contradicts that $f$ is an isomorphism (hence in particular injective).
Therefore, there is no such isomorphism $f$, thus the rings $2\Z$ and $3\Z$ are not isomorphic.

The Polynomial Rings $\Z[x]$ and $\Q[x]$ are Not Isomorphic
Prove that the rings $\Z[x]$ and $\Q[x]$ are not isomoprhic.
Proof.
We give three proofs.
The first two proofs use only the properties of ring homomorphism.
The third proof resort to the units of rings.
If you are familiar with units of $\Z[x]$, then the […]

A Maximal Ideal in the Ring of Continuous Functions and a Quotient Ring
Let $R$ be the ring of all continuous functions on the interval $[0, 2]$.
Let $I$ be the subset of $R$ defined by
\[I:=\{ f(x) \in R \mid f(1)=0\}.\]
Then prove that $I$ is an ideal of the ring $R$.
Moreover, show that $I$ is maximal and determine […]

Characteristic of an Integral Domain is 0 or a Prime Number
Let $R$ be a commutative ring with $1$. Show that if $R$ is an integral domain, then the characteristic of $R$ is either $0$ or a prime number $p$.
Definition of the characteristic of a ring.
The characteristic of a commutative ring $R$ with $1$ is defined as […]

The Quotient Ring by an Ideal of a Ring of Some Matrices is Isomorphic to $\Q$.
Let
\[R=\left\{\, \begin{bmatrix}
a & b\\
0& a
\end{bmatrix} \quad \middle | \quad a, b\in \Q \,\right\}.\]
Then the usual matrix addition and multiplication make $R$ an ring.
Let
\[J=\left\{\, \begin{bmatrix}
0 & b\\
0& 0
\end{bmatrix} […]

Generators of the Augmentation Ideal in a Group Ring
Let $R$ be a commutative ring with $1$ and let $G$ be a finite group with identity element $e$. Let $RG$ be the group ring. Then the map $\epsilon: RG \to R$ defined by
\[\epsilon(\sum_{i=1}^na_i g_i)=\sum_{i=1}^na_i,\]
where $a_i\in R$ and $G=\{g_i\}_{i=1}^n$, is a ring […]

$(x^3-y^2)$ is a Prime Ideal in the Ring $R[x, y]$, $R$ is an Integral Domain.
Let $R$ be an integral domain. Then prove that the ideal $(x^3-y^2)$ is a prime ideal in the ring $R[x, y]$.
Proof.
Consider the ring $R[t]$, where $t$ is a variable. Since $R$ is an integral domain, so is $R[t]$.
Define the function $\Psi:R[x,y] \to R[t]$ sending […]