Efficient Localization at a Prime Ideal Without Producing Unnecessary Primary Components

In Commutative Algebra, localization at a prime ideal in a polynomial ring is a basic but important tool. It is well-known that localization at a prime ideal can be computed through “primary decomposition”, however, it contains unnecessary primary components for the localization. In this paper, we propose a method for computing the localization without producing unnecessary primary components. Also, we discuss computation for desirable primary components from a view of “the degree of nilpotency”. In a computational experiment, we see the effectiveness of our method by its speciality.