Let R be an integral domain

(i) Let a belongs to R*\ U(R) (i.e a is non-zero, non-unit). Show that a is NOT an atom iff a = bc for proper divisors (b,c in R) of a

This is my attempt:

(==>) Suppose a is not an atom then: a is NOT in R*\ U(R), that is : 0 =(a) = R which implies a~0.

Also, a has proper divisor b| a <==> a = bc for b,c in R

Conversely, if a = bc (b,c in R) then b|a and so b~a.

Since a belongs to R*\ U(R), then a is non-zero non-unit which implies b is not in U(R). So a actually has a proper divisors

Thus, a is not an atom

Is the above argument correct?

(ii)prove that if a in R*\ U(R) then a is an atom <==> whenever a = bc for b,c in R, then b ~a or c~a

Proof:

Suppose that a is an atom and a = bc. Then b|a.Hence, b in U(R) or b ~a.

If b = ua for some u in U(R) then a =bc = uac= auc => uc =1 => c belongs to U(R)

Conversely, if a = bc then b~a or c~a. So, b and c belong to U(R). So, a = bc. Thus, a is an atom

Is that correct?

Please correct any mistakes you found on my work.

Thank you for taking your time