  The objective of this paper is to describe the structure of Zariski closed
algebras, which provide a useful generalization to finite dimensional algebras
in the study of representable algebras over finite fields. Our results include
a version of Wedderburn's principal theorem, as well as a more explicit
description using representations, in terms of "gluing" in Wedderburn
components. Finally, we construct "generic" Zariski closed algebras, whose
description is considerably more complicated than the description of generic
algebra of finite dimensional algebras.
  Special attention is given to infinite dimensional algebras over finite
fields.
