Question

Let R be an integral domain in which every ideal is principal. If(p)is a nonzero prime ideal in R, prove that p has this property: Whenever pfactors, p = cd , then c or d is a unit in R.

Short answer

It is proved that c or d is a unit of R.

Step-by-step solution

Statement of theorem 6.14

Theorem 6.14states that consider that as an ideal in a commutative ringwith identity. Then P would be a prime idealsuch that if the quotient ring R/P would be an integral domain.

Show that c or d is a unit in R

R/(p) would be an integral domain according to theorem 6.14 because (p) is a prime ideal.

Assume that p = cd, then there is $\left(c+\left(p\right)\right)\left(d+\left(p\right)\right)=0+\left(p\right)$ in R/(p). This implies that either $c\in \left(p\right)$or $d\in \left(p\right)$.

Assume that $c\in \left(p\right)$, therefore, there may be some $u\in R$ in which c = pu. then, there is p =pud and therefore, 1=ud by the cancellation. This implies that d is a unit.

Likewise, when $d\in \left(p\right)$, then c is a unit.

Hence, it is proved that c or dis a unit of R.