Unique factorization ___domain: Difference between revisions

Revision as of 00:03, 2 September 2024

Revision as of 17:23, 5 September 2024

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== Definition ==

Formally, a unique factorization ___domain is defined to be an [[integral ___domain]] ''R'' in which every non-zero element ''x'' of ''R'' which is not a unit can be written as a finite product of a [[Unit (ring theory)|unit]] ''u'' and zero or more [[irreducible element]]s ''p''_''i'' of ''R'':

: ''x'' = ''u'' ''p''₁ ''p''₂ ⋅⋅⋅ ''p''_''n'' with {{nowrap|''n'' ≥ 01}}

and this representation is unique in the following sense:

If ''q''₁, ..., ''q''_''m'' are irreducible elements of ''R'' and ''w'' is a unit such that

: ''x'' = ''w'' ''q''₁ ''q''₂ ⋅⋅⋅ ''q''_''m'' with {{nowrap|''m'' ≥ 01}},

then {{nowrap|1=''m'' = ''n''}}, and there exists a [[bijective|bijective map]] {{nowrap|''φ'' : {{mset|1, ..., ''n''}} → {{mset|1, ..., ''m''}}}} such that ''p''_''i'' is [[Unit_(ring_theory)#Associatedness|associated]] to ''q''_{''φ''(''i'')} for {{nowrap|''i'' ∈ {{mset|1, ..., ''n''}}}}.