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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) ...
View attachment 6410
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
View attachment 6410
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