Posts tagged ‘prime ideal’
I just explained how the affine plane could be described by the ring . A point M of the affine plane whose coordinate ring is R is a morphism defined by the assignment , where are the coordinates of M. In the case of points corresponding to morphisms , there is a natural way of recovering the point from the ring morphism by looking at his equations, which are elements of the kernel of the morphism. If M satisfies the equations and , then M has the form . This motivates the abstract definition of point of the affine plane as a morphism to some ring.
Conversely, the set of equations of M defines a canonically associated point , which is the morphism , where is the ideal generated by the equations. But this morphism has no reason to totally recover M if it wasn’t a point with integral coordinates. For example, the point is a special point, satisfying a lot of equations, which characterize it. But do not satisfy any polynomial equations with integral coefficients, so the set of its equations is empty, and cannot be used to recover it. Moreover, the point does not satisfy equations either: their algebraic properties are exactly the same. These points are called generic.
The prime spectrum of a ring is a convenient way of describing equivalence classes of points of a given ring.
Definition. The prime spectrum of is the set of points for prime ideals . If is any point of the affine plane with coordinates in an integral domain , then Mis canonically associated to some , where is the kernel of the map .