Question

Prove that every associate of an irreducible element is irreducible.

Short answer

It has been proved that every associate of an irreducible element is irreducible.

Step-by-step solution

Suppose that

Let p be an irreducible element of integral domain R.

Let q be its associate.

Then$q=pu$ for some unit u.

Prove that q is irreducible

Let q is reducible.

Then $q=ab$ where a and b are non units.

Then $pu=ab$.

This implies $p=\left(u^{-1}a\right)b$

Since p is irreducible therefore one of them must be unit.

Since bis non unit therefore$u^{-1}a$ must be unit.

Then $u\left(u^{-1}a\right)$ is a unit implying that a is a unit.

But this is a contradiction to the supposition.

Thus q is irreducible.

Conclusion

Thus it can be concluded every associate of an irreducible element is irreducible.