birkhoff's theorem from a geometric perspective: a simple example

‎from hilbert's theorem of zeroes‎, ‎and from noether's ideal theory‎, ‎birkhoff derived certain algebraic concepts (as explained by tholen) that have a dual significance in general toposes‎, ‎similar to their role in the original exles of algebraic geometry‎. ‎i will describe a simple exle that illustrates some of the aspects of this relationship‎. the dualization from algebra to geometry in the basic grothendieck spirit can be accomplished (without intervention of topological spaces) by the following method‎, ‎known as isbell conjugacy.