 For example,

An "R"-algebra "S" is called finitely generated (as an algebra) if there are finitely many elements "s"1, ..., "s""n" such that any element of "s" is expressible as a polynomial in the "s""i".Equivalently, "S" is isomorphic to

A much stronger condition is that "S" is finitely generated as an "R"-module, which means that any "s" can be expressed as a "R"-linear combination of some finite set "s"1, ..., "s""n".A ring is called local if it has only a single maximal ideal, denoted by "m".