Polinom ideallerinin Gröbner bazları kullanılarak primer bileşenlerine ayrılması

In this thesis, we investigate algorithms for computing primary decompositions of ideals in polynomial rings. Every ideal in a polynomial ring over a Noetherian commutative ring with identity has a primary decomposition, that is, it can be expressed as the intersection of primary ideals (in a unique way or not). The existence of primary decompositions in such polynomial rings is a result of the ascending chain condition and the existence proof does not suggest any construction method for the primary components of the ideal. The introduction of Gröbner basis by Buchberger has supplied powerful computational methods for various problems in commutative algebra. In the first part of the thesis, we investigate the algorithms developed by Gianni et al. \cite{Gianni88} for the computation of a primary decomposition of a given ideal in a polynomial ring. The main tool used in these algorithms is Gröbner basis techniques for the computation of certain operations on ideals. Properties of zero dimensional ideals are characterized in terms of Gröbner basis and based on that, an algorithm is developed for computing a primary decomposition of a zero dimensional ideal $I$ in a polynomial ring over a Noetherian domain $R$ such that $I \cap R$ is zero dimensional and primary. This algorithm is generalized to an algorithm for the computation of a primary decomposition of any given ideal in a polynomial ring over a PID. We give a complete discussion and analysis of the theorems and algorithms developed by Gianni et al. in \cite{Gianni88}. The second part of the thesis presents another approach to the computation of primary decomposition problem developed by Eisenbud et al. in \cite{Eisenbud92}. This method avoids the projection of an ideal to a polynomial subring with one less variable which was used for reduction in the algorithms developed by Gianni et al. We give an outline of the algorithms developed by Eisenbud et al. in \cite{Eisenbud92}. The algorithms developed by both Gianni et al. and Eisenbud et al. make it possible to compute primary components and associated primes of a given ideal, hence also the radical of the ideal. As a direct application of the computation of a primary decomposition of an ideal $I$ in a polynomial ring, the irreducible components of the variety of $I$ can be computed explicitly as the varieties of minimal associated primes of $I$.