# Definition:Irreducible Element of Ring

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## Definition

Let $\struct {D, +, \circ}$ be an integral domain whose zero is $0_D$.

Let $\struct {U_D, \circ}$ be the group of units of $\struct {D, +, \circ}$.

Let $x \in D: x \notin U_D, x \ne 0_D$, that is, $x$ is non-zero and not a unit.

### Definition 1

$x$ is defined as **irreducible** if and only if it has no non-trivial factorization in $D$.

That is, if and only if $x$ cannot be written as a product of two non-units.

### Definition 2

$x$ is defined as **irreducible** if and only if the only divisors of $x$ are its associates and the units of $D$.

That is, if and only if $x$ has no proper divisors.

## Also defined as

Some sources define the concept of an **irreducible element** only on an integral domain $\struct {D, +, \circ}$ which is also Euclidean.

## Also known as

An **irreducible element of a ring** is also known as an **atom of a ring**, hence described as **atomic**.

Some sources refer to such an element as a **prime element** of the **ring** in question.

## Also see

- Results about
**irreducible elements of rings**can be found**here**.