This paper continues our study of applications of factorized Gröbner basis computations in [8] and [9]. We describe a way to interweave factorized Gröbner bases and the ideas in [5] that leads to a significant speed up in the computation of isolated primes for well splitting examples. Based on that observation we generalize the algorithm presented in [22] to the computation of primary decompositions for modules. It rests on an ideal separation argument. We also discuss the practically important question how to extract a minimal primary decomposition, neither addressed in [5] nor in [17]. For that purpose we outline a method to detect necessary embedded primes in the output collection of our algorithm, similar to [22, cor. 2.22]. The algorithms are partly implemented in version 2.2.1 of our REDUCE package CALI [7].

Let k be a field, S = k[xv : v ϵ V] be the polynomial ring over the finite set of variables (xv : v ϵ V), and m = (xv : v ϵ V) the ideal defining the origin of Spec S. It is theoretically known (see e.g. Alonso et el., 1991) that the algorithmic ideas for the computation of ideal (and module) intersections, quotients, deciding radical membership etc. in S may be adopted not only for computations in the local ring Sm but also for term orders of mixed type with standard bases replacing Gröbner bases. Using the generalization of Mora's tangent cone algorithm to arbitrary term orders we give a detailed description of the necessary modifications and restrictions. In a second part we discuss a generalization of the deformation argument for standard bases and independent sets to term orders of mixed type. For local term orders these questions were investigated in Gräbe (1991). The main algorithmic ideas described are implemented in the author's REDUCE package CALI (Gräbe, 1993a).