Definition:Irreducible Element of Ring

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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.

Special cases