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