By definition, is a unit considered an irreducible?

[Werg22]

Just need a yes or no.

 

[Integral]

You need to define your terms.

 

[Werg22]

You need to define your terms.

I'm speaking of units and irreducibles in integral domains.

 

[Integral]

OK, so you have added an new term, and stil no definitions. ?

 

[mathwonk]

no, but one word answers are frustratingly not allowed here. so i had to enter it thrice.

 

[Werg22]

In an integral domain, a unit is an element that has an inverse in the integral domain, that is when they are multiplied together, they give 1.

An irreducible is an element of an integral domain that cannot be factored without one of its factor being a unit.

For example when the integral domain in question are the integers, the units are -1 and 1 and the irreducibles are the primes.

 

[Gib Z]

I would say your last line confirms the answer no, since -1 and 1 are not primes.

 

[morphism]

I think the confusion resulted from the definition of "irreducible". This adjective is exclusive to non-units (and nonzero elements!); see for example http://planetmath.org/encyclopedia/Irreducible.html .

 

[mathwonk]

yes werg omitted the part about an irreducible being a non unit.

in general think of how you want to understand a domain: you want to know how it differs from a field.
so first basic question: what are all the units?

second question: what are all the non units? hopefully there is some way to organize the non units, and one basic way is to define irreducibles, then hope to factor other elements into those.

so to undertstand non units we define irreducibles.

then we try to prove that under various simple hypotheses we can factor every non unit into irreducibles (e.g. noetherian domain)

then we hope for some uniqueness statement. obviously anything can be factored by any unit, so any uniqueness statement must akllow for this non uniqueness due to units.

sow ,e hope for a statement that factorization into irreducibles not only exists, but is unique except for an equivalence relation where multiplying by a unit is considered leaving things equivalent.

so we break the domain into two disjoint sets, units and non units. then the group of units acts on the multiplicatively closed set of non units. we consider elements of the same the equivalence class as "associates". then we try to factor non units into irreducibles, uniquely up to associates, which is not always possible.

i believe it is possible if the domain is noetherian, and it is possible to define gcd's of any two elements.

so a key concept is that of a gcd.