Irreducible element: Difference between revisions
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{{ | {{associate-invariant curing-element property}} | ||
==Definition== | ==Definition== | ||
A nonzero element in an [[integral domain]] is said to be '''irreducible''' if it is neither zero nor a unit, and it | ===In a commutative unital ring=== | ||
A nonzero element in a [[commutative unital ring]] is said to be '''irreducible''' if it is neither zero nor a unit, and given any factorization of the element as a product of two elements of the ring, it is [[defining ingredient::associate elements|associate]] to one of them. | |||
===In an integral domain=== | |||
In an [[integral domain]], there are two equivalent formulations. A nonzero element in an [[integral domain]] is said to be '''irreducible''' if it is neither zero nor a unit, and it satisfies the following equivalent conditions: | |||
* Any expression of it as a product of two elements has the property that one of the factors is [[associate elements|associate]] to it. | |||
* Any expression of it as a product of two elements has the property that one of the factors is a unit. | |||
==Relation with other properties== | ==Relation with other properties== | ||
Latest revision as of 22:28, 31 January 2009
Template:Associate-invariant curing-element property
Definition
In a commutative unital ring
A nonzero element in a commutative unital ring is said to be irreducible if it is neither zero nor a unit, and given any factorization of the element as a product of two elements of the ring, it is associate to one of them.
In an integral domain
In an integral domain, there are two equivalent formulations. A nonzero element in an integral domain is said to be irreducible if it is neither zero nor a unit, and it satisfies the following equivalent conditions:
- Any expression of it as a product of two elements has the property that one of the factors is associate to it.
- Any expression of it as a product of two elements has the property that one of the factors is a unit.
Relation with other properties
Stronger properties
- Prime element: For proof of the implication, refer Prime element implies irreducible and for proof of its strictness (i.e. the reverse implication being false) refer Irreducible element not implies prime
- Universal side divisor: For proof of the implication, refer Universal side divisor implies irreducible and for proof of its strictness (i.e. the reverse implication being false) refer Irreducible not implies universal side divisor