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

  1. $(A,+)$ is an abelian group

  2. $(A,\cdot)$ is a semigroup

  3. $\cdot$ distributes over $+$ on both sides

We only consider commutative rings with 1, that is, we require:

  1. $x\cdot y = y\cdot x$ for all $x, y \in R$

  2. 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. $1\in S$

  2. $x+y,x\cdot y,-x\in S$ for all $x,y \in S$

The latter condition is equivalent to requiring $x-y,x\cdot y \in S$ for all $x,y\in S$.

[In other contexts we do not require $1\in S$, but simply that $S$ is nonempty.]

Examples:

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

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

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

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

  5. 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 $R$:

    \[ R[x] = \{a_0 + a_1 x + ... + a_n x^n | n \ge 0, a_0, ..., a_n \in R\} \]