Hilbert's Nullstellensatz suggests an approach to algebraic geometry over any algebraically closed field "k": the maximal ideals in the polynomial ring "k"["x"1...,"x""n"] are in one-to-one correspondence with the set "k""n" of "n"-tuples of elements of "k", and the prime ideals correspond to the irreducible algebraic sets in "k""n", known as affine varieties. Motivated by these ideas, Emmy Noether and Wolfgang Krull developed the subject of commutative algebra in the 1920s and 1930s. Their work generalizes algebraic geometry in a purely algebraic direction: instead of studying the prime ideals in a polynomial ring, one can study the prime ideals in any commutative ring. For example, Krull defined the dimension of any commutative ring in terms of prime ideals. At least when the ring is Noetherian, he proved many of the properties one would want from the geometric notion of dimension.