Finite Algebra

A finite algebra over a commutative ring with multiplicative identity, in the commutative convention used here, is a unital commutative algebra over whose underlying module is a finitely generated module. Thus there are elements , ..., for which every element of is an -linear combination, with its module structure induced by the structure ring homomorphism (Milne 2014, p. 3). The structure ring homomorphism need not be injective. For example, is a finite algebra over , generated as a module by the class of 1.

Every finite algebra is a finitely generated algebra, and every element of it is integral over the base ring. Conversely, a finitely generated algebra is finite iff all of its elements are integral over the base ring (Stacks Project, Tag 00GH). For example, is finite over when is a monic polynomial of degree , since the classes of span it as a module. Finiteness refers to module generation, not to the number of elements of . Over a field, it is equivalent to finite vector space dimension.