Vol. 23, No. 3, 1967

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Algebras of global dimension one with a finite ideal lattice

William Edwin Clark

Vol. 23 (1967), No. 3, 463–471

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.

Mathematical Subject Classification
Primary: 16.50
Secondary: 18.00
Received: 31 March 1967
Published: 1 December 1967
William Edwin Clark