Let A denote a finite-dimensional
(associative) algebra over an algebraically closed field K. It is well known that A has
global dimension zero if and only if A is the direct sum of a finite number of full
matrix algebras over K. In this paper a specific representation is given for those
algebras A which have global dimension one (or less) and have only a finite number
of (two-sided) ideals. It is shown that every such algebra is isomorphic to a
(contracted) semigroup algebra K[S] over a subsemigroup S of the semigroup of all
n × n matrix units {eij}∪{0} which (i) contains e11,⋯,enn and (ii) contains eij or
eji whenever there are h and k such that ehi,eik and ehj,ejk are in S. Conversely, if
S satisfies (i) and (ii) then K[S] has global dimension one or less and has a finite
ideal lattice.
