Answer to Question #25259 in Abstract Algebra for Tsit Lam

Question

For any left ideal I in a ring R, define the core of I to be the sum of all ideals in I. Thus, core (I) is the (unique) largest ideal of R contained in I. Show that core (I) = ann(V ) where V is the left R-module R/I. (In particular, V is faithful iff core (I) = 0.)

Answer 1

First, ann(V ) is in I, and ann(V )is an ideal, so ann(V ) is in core(I). Secondly, core(I) · R is in core(I), that is inI, that implies core(I) · V =0, so core(I) is in ann(V ).