# A problem on $\text{ACCP}$

Let $R$ be a commutative ring. Could anyone advise me on how to prove $R$ has $\text{ACCP}$ (Ascending chain condition for principal ideals) iff every collection of principal ideals of $R$ has maximal element?

Hints will suffice, thank you.

• Thanks for great advice. Could you also advise me on how to prove if $R$ is an integral domain and has $\text{ACCP},$ then $R[X]$ has $\text{ACCP} \ ?$ – Alexy Vincenzo Mar 16 '14 at 6:19
• This means $R[X]$ is also an integral domain... – Alexy Vincenzo Mar 16 '14 at 6:22
• The integral domain hypothesis is important here. A principal ideal in R[X] is generated by a polynomial with coefficients in R. Note that since there are no zero divisors, the formula $deg(p\cdot q)=deg(p)+deg(q)$ holds. Hence given any principal ideal generated by a polynomial $p$ of degree $n$, there is a factorization that reduces the degree or not. If there is, keep going until the degree can't be made lower by factorization. If not, then all factorizations are just $r\cdot q$ for some $r\in R$, and you know that this eventually terminates because R satisfies ACCP. – Fred Byrd Mar 16 '14 at 6:35