A ring $R$ is a set with binary operations $+,\cdot$ such that

$(A,+)$ is an abelian group

$(A,\cdot)$ is a semigroup

$\cdot$ distributes over $+$ on both sides
We only consider commutative rings with 1, that is, we require:
$x\cdot y = y\cdot x$ for all $x, y \in R$
There exists $1\in R$ with $1\cdot x = x\cdot 1 = x$ for all $x \in R$
It is easily seen that if an identity element exists, it must be unique.
If $1 = 0$ then we must have $R = \{0\}$, which we call the zero ring.
A subset $S\subset R$ is a subring if:

$1\in S$

$x+y,x\cdot y,x\in S$ for all $x,y \in S$
The latter condition is equivalent to requiring $xy,x\cdot y \in S$ for all $x,y\in S$.
[In other contexts we do not require $1\in S$, but simply that $S$ is nonempty.]
Examples:

$\mathbb{Z}$ is the only subring of $\mathbb{Z}$.

$\mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$

The Gaussian integers $\mathbb{Z}[i]$ form a subring of $\mathbb{C}$.

$\mathbb{Z}_n$ is a ring. (Later we shall see this is the quotient ring $\mathbb{Z}/n\mathbb{Z}$.)

For any ring $R$, let
\[ R[[x]] =\{a_0 + a_1 x + a_2 x^2 + ...  a_0,a_1,a_2... \in R\} \]\[ R[x] = \{a_0 + a_1 x + ... + a_n x^n  n \ge 0, a_0, ..., a_n \in R\} \]This is the formal power series over $R$. One subring of this is the ring of polynomials over $R$: