Trace problem

[Kurt Foster <drsardonicus@earthlink.net>]

Let K = bnfinit(T)) (T monic and irreducuble in Z[x]) be a number  
field, R = K.zk its ring of integers, x an element of R.  I want to  
determine all units u such that trace(Mod(x*u,T)) is zero.

It's certainly possible to obtain a Z-basis of the set of y in R for  
which trace(Mod(x*y, T)) is zero.  But how to determine which units  
(if any) are in the Z-module spanned by that basis has me flummoxed.   
Perhaps I am overlooking something obvious.

I suppose you could reduce modulo a rational integer m and look at the  
units mod mR.  That might lead to a proof of nonexistence if no such  
units exist.  Perhaps localization would help?