Category:Definitions/Irreducible Elements of Rings

This category contains definitions related to Irreducible Elements of Rings.
Related results can be found in Category: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.