Are units considered irreducible

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In the context of integral domains, units are not considered irreducible elements. An element r in an integral domain R is defined as irreducible if it cannot be expressed as a product of two non-unit elements. The discussion clarifies that if v is a unit in R, then any factorization of v into ab implies that at least one of a or b must be a unit, thus confirming that units do not meet the criteria for irreducibility.

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Definition: Let R be an integral domain. A nonzero, nonunit element r in R is said to be irreducible if whenever r=ab, then a is a unit or b is a unit.

My question is are units considered irreducible.

This how I understand it,
Let v in R be a unit such that v=ab ==> 1=ab(v^-1) ==> 1=a[b(v^1)] ==> a is unit.
So according to this, v is irreducible.
Am I right? Help!
Thanks.
 
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Huh? Is the question about which word was being defined in the definition?
 
no. units are not irreducibles by definition.
 

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