Commutative Algebra

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:

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

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

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

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

   $$R[[x]]=\{a_0+a_{1}x+a_2{x}^2+...\mid a_0,a_1,a_2...\in R\}$$

   This is the formal power series over $R$. One subring of this is the ring of polynomials over $R$:

   $$R[x]=\{a_0+a_{1}x+...+a_n{x}^n\mid n\ge 0,a_0,...,a_n\in R\}$$