# Introduction

A basic algebra is a finite dimensional algebra $A$ over a field, all of whose simple modules have dimension one. In the literature such an algebra is known as a “split” basic algebra. Every algebra is Morita equivalent to a basic algebra, though a field extension may be necessary to obtain the split basic algebra. Magma has several functions that create the basic algebras corresponding to algebras of different types.

The type `AlgBas` in Magma is optimized for the purposes of doing homological calculations. A basic algebra $A$ is generated by elements $a_1, a_2, \ldots,  a_t$ where $a_1, \ldots, a_s$ are the primitive idempotent generators and $a_{s+1}, \ldots, a_t$ are the nonidempotent generators. Each nonidempotent generator, $a_k$ must have the property that $a_i*a_k*a_j = a_k$ for specific idempotent generators $a_i$ and $a_j$. The projective indecomposable modules have the form $P_i = a_i \cdot A$ for $i = 1, \ldots, s$ and the simple modules have the form $S_i = P_i/\hbox{Rad}(P_i)$, where $\hbox{Rad}(P_i)$ is the radical of $P_i$.