Studies in Rings generalised Unique Factorisation Rings

-15-
are distinguished by many nice propertieso
For instance,
every vector space is a direct sum of one-dimensional
subspaces. We view simple modules as analogous to one
dimensional spaces, :and
the corresponding analogoues to
higher dimenaional vector spaces are the semisimple
modules; modules which are direct sums of simple submodules.
Definition 1.32 0
The
socle of an R-module A is the sum of all simple
submodules of A and is denoted by sac A. A is sernisimple
i"f A = s oc A.
In any ring R, it is easy to observe that soc (RR)
is an ideal of R. Similarly soc (RR) is an ideal of R,
but these two socles need not coincide in generalo However,
there are rings in which these two coincide. For instance
R = Mn(D),
where n is a positive integer and 0 is a division
ring. In case n = 2, the right idea Is 1 1
= [g gJ 1 2
= [g gJ
are the simple right ideals and M 2(D)
= 1 1
fB 1 2
• Similarly
M 2 (D) = J l $ 3 2, where 3 1
= m~ J 2 = [g g] are the
simple left ideals. We state a propositiono