Rings

Introduction

We’ve introduced groups as an abstract structure that contains a set of elements together with a binary operation, whereas some specific group axioms have to be satisfied. On the contrary to a group $G$, a ring $R$ contains two mathematical operations: an addition $+$ and a multiplication $\cdot$. A ring is defined as an abstract structure that maintains an abelian group structure under the addition operation but not necessarily under multiplication.

Definition of a ring

A set of elements $R$ together with both operations

$$\begin{aligned} &+ \space : R \times R \rightarrow R,(a, b) \mapsto a+b, \quad \text { and } \\ &\cdot \space: R \times R \rightarrow R,(a, b) \mapsto a \cdot b \end{aligned}$$

is called a ring, if the following properties are fulfilled:

• R1: $R$ is an abelian group under addition $(+)$.
• R2: The multiplication $\cdot$ is associative.
• R3: The multiplication $(\cdot)$ is distributive with respect to the addition $(+)$, i.e., for all $a, b, c \in R$ :

  $$a \cdot(b+c)=a \cdot b+a \cdot c \text { and }(a+b) \cdot c=a \cdot c+b \cdot c.$$

The neutral element $0 \in R$ of the addition is called the zero elements of $R$.

Note: We usually write $(R,+, \cdot)$ to $R$. The requirements for multiplication are weaker than the ones for addition because the multiplication operation doesn’t have to be necessarily commutative. There’s also no requirement for the existence of a multiplicative neutral element $1$.