# Category:Irreducible Elements of Rings

This category contains results about Irreducible Elements of Rings.

Definitions specific to this category can be found in Definitions/Irreducible Elements of Rings.

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.

## Pages in category "Irreducible Elements of Rings"

This category contains only the following page.