If R is noetherian then spec R is noetherian. (In this case noetherian refers to the ideals of R, not left or right ideals.) A descending chain of closed sets would imply an ascending chain of ideals, which is impossible.
don't be confused about the term noetherian as it is applied to spec R. A topology is noetherian if it has no infinite descending chains of closed sets. In every other context, noetherian refers to ascending chains, but not here. Why the switch? So that R noetherian implies spec R noetherian. That's all.
Similarly, R artinian implies spec R is artinian, with no infinite ascending chains of closed sets.
Don't assume the converse. Let R be Q (the rationals) adjoin infinitely many commuting indeterminants x1 x2 x3 …, such that each xi raised to the fifth power is 0. Every indeterminant is nilpotent, and belongs to the single prime ideal P, which is of course maximal.
Bring in or take out indeterminants one at a time, and build an ascending or descending chain of ideals inside P, hence R is neither noetherian nor artinian. Yet spec R is the simplest possible space, consisting of one point.