Category:Irreducible Elements of Rings

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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.