A prime, noetherian, hereditary ring \(R\) whose nonzero ideals are invertible is called a Dedekind prime ring. It is well-known that a commutative domain is Dedekind if (and only if) its ideals are products of prime ideals. Due to \textit{A. W. Chatters} and \textit{C. R. Hajarnavis} [J. Algebra 122, 475-480 (1989; Zbl 0671.16004)], the ``if'' part remains true when \(R\) is a noetherian prime PI ring. Since the noetherian assumption is not needed to establish this in the commutative case, the question arose whether it is also dispensable in the PI case, at least in special circumstances. The present paper deals with this, showing that if \(R\) is a prime ring which is either module finite over its center, or an affine PI ring, and if every ideal of \(R\) is the product of prime ideals, then \(R\) is noetherian, and hence a Dedekind prime ring. The proof is an interesting mix of clever elementary ideal arithmetic and the use of some deep results from PI theory, such as, for example, the Artin-Procesi Theorem, Cauchon's theorem that a semiprime PI ring with ACC for ideals is noetherian, and Schelter's result on the catenarity of affine PI rings.