Bachelor of Science

Category theory provides a common framework for many branches of mathematics, especially topology and algebra. In this thesis, we mainly deal with those categories in algebra. Locally, the study of category theory helps in the understanding of the essence of important concepts in algebra such as injective mappings, subjective mappings, isomorphisms, identity elements and products, etc. Globally, it gives a way of comparing categories that used to be regarded as absolutely different things. The thesis yields a deeper understanding of the categories arising in algebra through the application of fundamental concepts in category theory, namely functors, natural transformations and adjoint functors.

Given any vestor space V, the operators on V form an algebra R. In this project, we investigate the structures of left (right) ideals of such algebras. It can be shown that, when V is M e dimensional, there is a one-to-one correspondence between subspaces of V and left (right) ideals of V. We also proved that in a Hilbert space, a subspace is determined by a left ideal if and only if it is closed.