#### Faculty Advisor

Keane, Patrick Gerard

#### Abstract

An algebra over a field (simply “algebra” for short) is an algebraic structure consisting of a vector space augmented with a vector multiplication operation obeying suitable axioms. Algebras are well behaved and have notions of dimension, basis, subalgebras, algebra ideals, algebra homomorphisms, and quotient algebras largely analogous to those of vector spaces or rings. Algebras occur often in mathematics, for example the set of all n×n matrices valued in a field k form an algebra M_n(k). We investigate which integers occur as dimensions of subalgebras of M_n(k). We give a description of the dimensions of simple, nilpotent, and semisimple matrix subalgebras along with several sequences that represent various properties of matrix subalgebras.

#### Publisher

Worcester Polytechnic Institute

#### Date Accepted

April 2019

#### Major

Mathematical Sciences

#### Project Type

Major Qualifying Project

#### Copyright Statement

All authors have granted to WPI a nonexclusive royalty-free license to distribute copies of the work, subject to other agreements. Copyright is held by the author or authors, with all rights reserved, unless otherwise noted.

#### Accessibility

Unrestricted

#### Advisor Department

Mathematical Sciences