# Rings

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 $$R$$:

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

Ben Lynn blynn@cs.stanford.edu 💡