This is a paper on Noetherian rings from a constructive point of view. It is a natural continuation of a previous paper by the author [Trans. Am. Math. Soc. 197, 273-313 (1974; Zbl 0356.13007)] where it was shown how to construct a primary decomposition and to find the associated prime ideals of a given ideal in a polynomial ring over a field. - In another paper [Rend. Semin. Mat. Fis. Milano 44 (1974), 55-61 (1975; Zbl 0345.13010)], the author defined a ring R to be Noetherian if the following two conditions are satisfied: (a) Given any chain \(A_ 1\subseteq A_ 2\subseteq..\). of finitely generated ideals in R (via finite sets of generators), one can find an i such that \(A_ i=A_{i+1}\). (b) Given any two finitely generated ideals A and B (via finite sets of generators), one can construct a finite set of generators for \(A\cap B\) and A:B. It is easy to see that the condition (a) already implies the existence of a j such that \(A_ k=A_ j\) for every \(k\geq j\), so that R is also Noetherian from the classical point of view. In the previously cited paper the author showed that the above defining conditions on R transfer constructively to R[X]. In the present paper, the author considers the following question: Suppose that for any given ideal (via a finite set of generators) of the Noetherian ring R one is given a normal decomposition of it into primary ideals together with the associated primes, or one can construct one, then can one construct the same for any given ideal of R[X]? - He shows that if R is a Noetherian ring in which one can compute and in which every finitely generated ideal A is detachable (i.e. given a in R, one can tell whether a is in A), then the question has an affirmative answer if and only if the following two conditions are satisfied: (F) If p is any prime ideal of R, then any non zero polynomial in \((R_ p/pR_ p)[X]\) can be effectively factorized into irreducible factors. (P) If p is any prime ideal of R, if q is the characteristic of the field \(R_ p/pR_ p\), then any elements \(z_ 1,...,z_ s\) of \(R_ p/pR_ p\) can be checked for q-independence, and in the case they are q-independent, one can construct an equation exhibiting this.