Definition:Irreducible (Ring Theory)

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Definition

Let $\left({D, +, \circ}\right)$ be an integral domain whose zero is $0_D$.

Let $\left({U_D, \circ}\right)$ be the group of units of $\left({D, +, \circ}\right)$.

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

Then $x$ is defined as irreducible iff it has no non-trivial factorization in $D$.

Some sources call such an element an atom.

The definition can alternatively be stated:

$x$ is irreducible iff the only divisors of $x$ are its associates and the units of $D$.
$x$ is irreducible iff it has no proper divisors.
$x$ is irreducible iff it cannot be written as a product of two non-units.

Some sources define the concept of irreducibility only when an integral domain $\left({D, +, \circ}\right)$ is Euclidean.

Irreducible elements of the Ring of Polynomial Functions play an important role in the Galois theory of fields.

By Units of Ring of Polynomial Forms over a Field, a polynomial in a single indeterminate with coeffifients in a field is irreducible if and only if it is not a product of two polynomials of smaller degree.

This is not necessarily true for polynomials over a commutative ring.