- #1
Math Amateur
Gold Member
MHB
- 3,997
- 48
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) ...
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
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