Joseph A. Gallian, in his book, "Contemporary Abstract Algebra" (Fifth Edition) defines an irreducible element in a domain as follows ... (he also defines associates and primes but I'm focused on irreducibles) ...(adsbygoogle = window.adsbygoogle || []).push({});

I am trying to get a good sense of this definition ...

My questions are as follows:

(1) Why are we dealing with a definition restricted to an integral domain ... why can't we deal with a general ring ... presumably we don't want zero divisors ... but why ...

(2) What is the logic or rationale for excluding a unit ...that is why is a unit not allowed to be an irreducible element ..

(3) We read that for an irreducible element ##a##, if ##a = bc## then ##b## or ##c## is a unit ... ... why is this ... ... ? ... ... ... presumably for an irreducible we want to avoid a situation where ##a## has a "genuine" factorisation ... but how does ##b## or ##c## being a unit achieve this ...

Hope someone can help ...

Peter

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# I Definition of an irreducible element in an integral domain

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