prime and atom

**knguyen2005** · October 30th 2009, 09:07 AM

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

**proscientia** · October 30th 2009, 04:50 PM

$a\sim b\ \Leftrightarrow\ a=ub$ 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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