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 RP 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 RP 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 RP, 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.