Operations on ideals in polynomial rings
Christofer Östlin · KTH Publication Database DiVA (KTH Royal Institute of Technology) · 2014
A ring is an algebraic structure equipped with two binary operations satisfying certain axioms, providing it with specific and highly useful properties. Furthermore, letting the elements of such structure constitute the coefficients in polynomials forms a new set in which the ring properties are preserved – a polynomial ring. To gain a deeper understanding of the rings, one may consider additively closed and multiplicatively absorbing subsets known as ideals. In the case of polynomial rings, ideals are of particular importance as they often are used to define field extensions to introduce solutions to polynomial equations, one such example being the complex numbers C. Ideals can be explicitly defined by means of generating sets; a number of key elements that, when applying the rules of operation to them, generates the entire ideal. Ideals generated solely by single-termed polynomials (monomials) in two or three variables were the focus of this thesis. We outlined a method to graphically visualize these monomial ideals and examined how it could facilitate operations over them. More specifically, we explored basic arithmetics and found that adding two monomial ideals is equivalent to superimposing their graphical representations, and that multiplying them corresponded to point vector addition between the figures. Additionally, we considered an algebraic approach to finding the integral closure of a monomial ideal and derived a graphically analog method generalizable to higher dimensions. Lastly, we hypothesized how the visualizations could be further utilized to aid in identifying minimal monomial reductions and suggested the incorporation of ideas from linear algebra to further advance such procedure. Bachelor thesis Department of Mathematics, Uppsala University Sweden, 2014