# Basic Definitions

• Definition of a commutative ring

A ring $R$ is a commutative ring if $ab=ba$ for all $a, b \in R$.

• Definition of an integral domain

An integral domain is a commutative ring $R$ with $0 \neq 1$ such that whenever $ab=0$ then either $a = 0$ or $b=0$.

• Definition of a field

An field is a commutative ring $F$ with $0 \neq 1$ such that every nonzero element $a \in F$ has an inverse in $F$, that is, there is an element $a^{-1} \in F$ such that $aa^{-1} = 1 = a^{-1}a.$

• Definition of a subring

A subset $S$ of a ring $R$ is a subring of $R$ if

(1) $1 \in S;$

(2) $a - b \in S$ for all $a, b \in S;$

(3) $ab \in S$ for all $a, b \in S.$

• Definition of a subfield

A subring $S$ of a field $F$ is a subfield of $F$ if every nonzero element of $S$ has an inverse in $S.$

• Definition of characteristic

Let $R$ be a ring. If there is a positive integer $n$ such that $n1 = 0$, then the least such integer is called the characteristic of $R$. If no such integer exists, then we say the characteristic of $R$ is zero.

• Theorem

Let $R$ be an integral domain. Then the characteristic of $R$ is either zero or a prime.

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