Localization, Zero is a Local Property

Local Property

The term locally is an adverb that describes a property of a ring, rather than the ring itself. For instance, imagine a ring can be green. We say that the ring R is locally green if every localization R_P is green. Furthermore, green is a local property if the following holds. A ring is green iff all its localizations are green. In other words, a ring is green iff it is locally green.

These concepts apply to ideals, and even elements of the ring. For instance, an element x is locally green iff all its localizations x/1 in R_P are green.

Zero is a Local Property

An element x is 0 iff all its localizations are 0.

Since 0/1 represents the zero element in each R_P, one direction is obvious. We only need show the converse. And for this we need R to be commutative.

Let J be the ideal in R that kills x. Since 1 does not kill x, J is a proper ideal, and embeds in a maximal ideal M, which happens to be prime. Localize about M, and x/1 is equivalent to 0/1. In other words, ux = 0 for some u outside of M. Since all the elements that kill x lie inside M, we have a contradiction. Therefore x is 0 iff it is locally 0.

More specifically, x = 0 iff its localizations about maximal ideals are all zero. This is not unusual. A local property is often local with respect to maximal ideals.