This paper deals with the
relationship between a ring T and the idealizer R of a right ideal M of T. [The ring R
is the largest subring of T which contains M as a two-sided ideal.] Assuming M to be
a finite intersection of maximal right ideals of T, the properties of T and R are
shown to be very similar. The main theorem of the first section shows that
under these hypotheses the right global dimensions of T and R almost always
coincide. In the second section, where T is assumed to be a nonsingular ring,
the maior theorem asserts that the singular submodule of every R-module
is a direct summand if and only if the corresponding property holds for
T-modules.
