Prove or disprove if D is an integral domain then every prim

Prove or disprove: if D is an integral domain, then every prime element in D is also irreducible in D.

if D is an integral domain, then every prime element in D is also irreducible in D the statement is false.

let D is a integral domain.

and let a D is prime element

now assume that Assume a = mn in D

so a|mn implies either a|m or a|n as a is prime

let a|m than m = aD

we have a = mn so a = anD hence nD = 1 since we’re in an integral domain and n is a unit.

so in general, irreducibles might not be prime.

or other proof like:

Suppose that D is an integral domain, and suppose p is an element that is reducible as p = ab, with neither a nor b a unit.

Then p divides itself, the product of a and b, but does not divide either a or b

if it divided one, the other must be a unit as multiplicative cancellation holds in an integral domain.

Therefore p cannot be prime.