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
$\displaystyle a\sim b\ \Leftrightarrow\ a=ub$ where $\displaystyle u\in R$ is a unit. As $\displaystyle u$ is a unit, $\displaystyle uv=1$ for some $\displaystyle v\in R$ and so $\displaystyle b=va.$
(i)
Suppose $\displaystyle a$ is prime and $\displaystyle b\mid cd$ $\displaystyle (c,d\in R).$ Then $\displaystyle cd=be=a(ve)$ for some $\displaystyle e\in R.$ Hence $\displaystyle a\mid cd$ $\displaystyle \implies$ $\displaystyle a\mid c$ or $\displaystyle a\mid d$ as $\displaystyle a$ is prime. So either $\displaystyle fa=c$ or $\displaystyle ga=d$ for some $\displaystyle f,g\in R$ $\displaystyle \implies$ either $\displaystyle (uf)b=c$ or $\displaystyle (ug)b=d$ $\displaystyle \implies$ $\displaystyle b\mid c$ or $\displaystyle b\mid d$ $\displaystyle \implies$ $\displaystyle b$ is prime.
The other implication follows by interchanging $\displaystyle a$ and $\displaystyle b$ and interchanging $\displaystyle u$ and $\displaystyle v.$
(ii)
Suppose $\displaystyle a$ is irreducible and let $\displaystyle b=cd$ for some $\displaystyle c,d\in R.$ Then $\displaystyle a=(uc)d$ and since $\displaystyle a$ is irreducible, either $\displaystyle uc$ or $\displaystyle d$ is a unit. If $\displaystyle uc$ is a unit, then $\displaystyle c=v(uc)$ is a unit. Thus $\displaystyle b=cd\ \Rightarrow\ c\ \mbox{or}\ d$ is a unit, proving that $\displaystyle b$ is irreducible.
Again the other implication follows by swapping $\displaystyle a\,/\,u$ and $\displaystyle b\,/\,v.$