Positive answers to analogues of B�chi's Problem in various other rings of functions have been obtained since then ( in the case of rings of functions, one adds the hypothesis that not all " x " " n " are constant ).
12.
The case of an algebraically closed ground field is specially important is algebraic geometry, because, in this case, the above homeomorphism is a homeomorphism between the affine space and the set of all maximal ideals of the ring of functions ( this is Hilbert's Nullstellensatz ).
13.
While the theory has developed in its own right, a fairly recent trend has sought to parallel the commutative development by building the theory of certain classes of noncommutative rings in a geometric fashion as if they were rings of functions on ( non-existent )'noncommutative spaces '.
