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
where is a unit. As is a unit, for some and so
(i)
Suppose is prime and Then for some Hence or as is prime. So either or for some either or or is prime.
The other implication follows by interchanging and and interchanging and
(ii)
Suppose is irreducible and let for some Then and since is irreducible, either or is a unit. If is a unit, then is a unit. Thus is a unit, proving that is irreducible.
Again the other implication follows by swapping and