Unit

An element $v$ of $D$ having a multiplicative inverse in $D$ is called a unit (regular element) of $D$.

Associate

An element $b$ of $D$ is called an associate of $a\in D$ if $b=v\cdot a$, where $v$ is some unit of $D$.

Divisor

An element $a$ of $D$ is a divisor of $b\in D$ provided there exists an element $c$ of $D$ such that $b=a\cdot c$.

Note: Every non-zero element $b$ of $D$ has as divisors its associates in $D$ and the units of $D$. These divisors are called trivial (improper); all other divisors, if any, are called non-trivial (proper).

Irreducible

An element $a$ in an integral domain $R$ is called irreducible if it is not zero or a unit, and if whenever $a$ is written as the product of two elements of $R$, one of these is a unit.

Prime

An element $p$ of an integral domain $R$ is called prime if $p$ is not zero or a unit, and whenever $p$ divides $ab$ for elements $a$, $b$ of $R$, either $p$ divides $a$ or $p$ divides $b$.

Note that if $R$ is an integral domain, every prime element of $R$ is irreducible.

Associates

Elements $r$ and $s$ are called associates of each other if $s=ur$ for a unit $u$ of $R$. So $a\in R$ is irreducible if it can only be factorized as the product of a unit and one of its own associates.

Unique Factorization Domain

An integral domain $R$ is a unique factorization domain if the following conditions hold for each element $a$ of $R$ that is neither zero nor a unit:
i) $a$ can be written as the product of a finite number of irreducible elements of $R$.
ii) This can be done in an essentially unique way. If $a=p_1{p}_2\cdots p_r$ and $a=q_1{q}_2\cdots q_s$ are two expressions for $a$ as a product of irreducible elements, then $s=r$ and $q_1,\cdots ,q_s$ can be reordered so that for each $i$, $q_i$ is an associate of $p_i$.