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Ring

Definition

A ring $(R,\times,+)$ is a set equipped with two binary operations such that.

• $(R,+)$ is an abelian group, whose identity is denoted $0$.
• $\times$ is associative.
• There is a multiplicative identity $1 \in R$ s.t. $r \times 1 = r = 1 \times r \forall r \in R$.
• Multiplication is distributive over addition, i.e. $a\times(b+c)=a\times b + a \times c$ and $(a+b)\times c=a\times c + b \times c$ for all $a,b,c \in R$.

Properties

Anything multiplied by zero is zero: let $a \in R$, then $0a = (0-0)a = 0a - 0a = 0$ holds by the distributiivity laws.

The trivial ring $\{0\}$ is the only ring such that $0 = 1$. Proof: suppose $0=1$ in $R$ and $a \in R$. Then $a = 1a = 0a = 0$.

Element types

If $r, s \in R\setminus\{0\}$ s.t. $sr=0$ then we say $r,s$ are zero-divisors.

Types

If multiplication is commutative, we say R is commutative.

If R is commutative and has no zero-divisors, we say R is an integral domain.

If all ideals of $R$ are principal, we say R is a principal ideal domain.

Fields are Euclidean domains are principal ideal domains are unique factorisation domains are integral domains are commutative rings are rings.

Examples

$\mathbb{Z}, \mathbb{F}[t]$ are a Euclidean domains.

The set of linear transformations $T: V \to V$, and more generally the set of endomorphisms of a module, is a non-commutative ring.

The set of continuous functions $\mathbb{R} \to \mathbb{R}$ (with pointwise addition & multiplication) is a commutative ring but not an integral domain. The set of holomorphic functions $U \to \mathbb{C}$ (where $U$ is an open subset of $\mathbb{C}$) is an integral domain.

all these pages adapted with probably insufficient credit from my university's lecture notes