Algebraic Geometry
Spaces to Rings and Back Again!
 Given any space X, the set R(X, k) of maps from X to a field k
is itself a ring.
 Given a ring A, the set Spec(A) of prime ideals of A is a space.
 In fact, if you set things up right, R(Spec(A), k) = A and
Spec(R(X, k)) = X.
In Algebraic Geometry, we study spaces by their rings and rings by their spaces!
Examples
 The equation y^{2} = x^{3} + x^{2}  2x defines an elliptic curve in the real plane.
 You get more solutions over the complex numbers.
 Here's what the rest of it looks
like over the complex numbers.
 The ring you get from the elliptic curve is C[x,y]/(y^{2}  x^{3}  x^{2} + 2x); the space you get from C[x,y]/(y^{2}  x^{3}  x^{2} + 2x) is the elliptic curve!
For more, see:
 B. Harbourne, Seshadri constants and very ample divisors on algebraic surfaces, J. Reine Angew. Math. 559 (2003) 115122.
 A. Seceleanu, Inverse systems, fat points and the weak Lefschetz property, joint with B. Harbourne and H. Schenck, J. Lond. Math. Soc. (2) 84 (2011), no. 3, 712730.

J. Walker, Codes and Curves, Amer. Math. Soc. book, 2000
 S. Wiegand, Projective lines over onedimensional semilocal domains and spectra of birational extensions, in: Algebra Geometry and Applications, (C. Bajaj, ed.), Springer Verlag, New York 1994.
 W. Zhang, Lyubeznik numbers of projective schemes, Adv. Math. 228 (2011), 575616.