4.2 Rings and Fields

Definitions

We are going to define a ring $(R+\cdot )$ or simply $R$.

Data:

• A set $R$ of elements.

• An operation $+:R\times R\to R$ called addition.

• An operation $\cdot :R\times R\to R$ called multiplication.

Axioms:

• $(R,+)$is an abelian group.

• Multiplication is left and right distributive over addition:

• $a(b+c)=ab+ac$ and$(a+b)c=ac+bc$ for all $abc\in R$

• Multiplication is associative: $(ab)c=a(bc)$ for all $abc\in R$.

• Multiplication has a neutral element denoted by $1$.

In this case, we say that $(R,+,\cdot )$ is an associative ring with an identity. We will use the term ring to denote an associative ring with an identity. As before, we usually say that $R$ is a ring assuming that the operations in use are clear. The neutral element with respect to addition is denoted by $0$ and is called zero. If $a\in R$ is an arbitrary element, its inverse with respect to addition is denoted by $-a$. For any $a,b\in R$, the expression $a+(-b)$ will be denoted by $a-b$ for short. If in addition we have

• Multiplication is commutative: $ab=ba$ for all $ab\in R$.

The ring is said to be commutative.

Field

A field is a set $F$ which is closed under two operations $+$ and $\times$ such that:

• $F$is an abelian group under$+$

• $F-\left\{0\right\}$ the set $F$ without the additive identity$0$is an abelian group under$\times$

Alternative Definition: A field is a commutative ring where every non-zero element is invertible with respect to multiplication, and $1\ne 0$. More explicitly, if we add the following conditions to a commutative ring:

• Every non-zero element is invertible with respect to multiplication: for every $a\in R\setminus \left\{0\right\}$, there exists an element $b\in R$ such that $ab=ba=1$.

• $1\ne 0$

The ring is said to be a field. In this case, the inverse element for $a$ is denoted by $a^{-1}$.

Subring

Let $R$ be a ring. We are going to define a subring $T\subseteq R$.

Data:

• A subset $T\subseteq R$.

Axioms:

• $(T,+)\subseteq (R,+)$ is a subgroup.

• $T$ is closed under multiplication.

• $T$ contains$1$

Ideals

Suppose that $(R+\cdot )$ is a ring. I am going to define an ideal $I$ in the ring $R$.

Data:

• A subset $I\subseteq R$.

Axioms:

• $(I,+)\subseteq (R,+)$is a subgroup.

• For any $r\in R$ we have

  $rI=\{rx|x\in I\}\subseteq I$ and $Ir=\{xr|x\in I\}\subseteq I$

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