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Math Help - prime and atom

  1. #1
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    prime and atom

    Suppose that a,b in R, where R is an integral domainand a~b.
    Show that: (i) a is prime if and only if b is prime
    (ii) a is an atom if and only if b is an atom
    how do you prove it?

    Thank you very much
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  2. #2
    Junior Member
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    Oct 2009
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    a\sim b\ \Leftrightarrow\ a=ub where u\in R is a unit. As u is a unit, uv=1 for some v\in R and so b=va.


    (i)

    Suppose a is prime and b\mid cd (c,d\in R). Then cd=be=a(ve) for some e\in R. Hence a\mid cd \implies a\mid c or a\mid d as a is prime. So either fa=c or ga=d for some f,g\in R \implies either (uf)b=c or (ug)b=d \implies b\mid c or b\mid d \implies b is prime.

    The other implication follows by interchanging a and b and interchanging u and v.


    (ii)

    Suppose a is irreducible and let b=cd for some c,d\in R. Then a=(uc)d and since a is irreducible, either uc or d is a unit. If uc is a unit, then c=v(uc) is a unit. Thus b=cd\ \Rightarrow\ c\ \mbox{or}\ d is a unit, proving that b is irreducible.

    Again the other implication follows by swapping a\,/\,u and b\,/\,v.
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