The condition for integral domain R to be P.I.D. If $a_{i}$ are nonzero element in R such that $a_{i+1}$ divides $a_{i}$ for all i, then there is a positive integer N s.t $a_{n}$ is a unit times $a_{N}$ for any $n$ larger than N.
I wonder that given condition imply "every ideal of R is finitely generated"
How to check this?

You mention the ascending chain condition for principal ideals. 
This condition, together with the condition that ideals generated by irreducible elements are  prime, characterises unique factorisation domains (UFDs).
Now not all UFDs are noetherian. E.g. the ring of polynomials in a countable number of indeterminates over a field $K$, $K[X_n]_{n\in\mathbf N}$, while being a UFD, is not noetherian.
