Summary of lecture 2
			Simple and semisimple modules    
			-----------------------------

simple module (definition):  a module that admits exactly two submodules: 0 and itself.
	Note that the zero module is not considered simple.
	This is like 1 not being defined to be prime.  

Schur's lemma:   version 1.   Any non-zero module map from a simple module is an injection.
        Any non-zero module map to a simple module is a surjection.
	Any non-zero map between simple modules is a bijection.
	The endomorphism ring of a simple module is a division ring.

	version 2.  Let M be a simple finite dimensional module for some
	k-algebra where k is an algberaically closed field.   Then the only
	endomorphisms of M are the scalar mulitplications (by elements of k).

Semisimple module (defn):  Three equivalent conditions for a module M over a ring
	1. M is the sum of all its simple submodules.
	2. M is a direct sum of simple submodules.
	3. Every submodule N of M has a complement, that is, submodule P such
	that M is the (internal) direct sum of N and P.

Proof given.  But the proof of 3 implies 2 was given only in the case when M
is a finite dimensional module over a k-algebra.   The general case is outlined
in one of the exercises.

	Corollary:  Subs and quotients of semisimple modules are semisimple.

Uniqueness of decomposition of a semisimple module as a direct sum of simples:
	Notion of S-isotypic component (where S is an isomorphism class of
	simple modules):  sum of images of all module maps from to S to M
	Any semisimple module is the direct sum, as S varies over all
	isomorphim classes of simple modules,  of its S-isotypic components.

	The uniqueness of multiplicities stated and proved only in the
	case of finite dimensional modules:    multiplicity of S in the
	S-isotypic component is the ratio of the dimension (as a vector space)
	of the isotypic component to that of S.

Structure of the endomorphism ring of a semisimple module:
	First let M be a semisimple module over any ring R and
	suppose that I_1 + I_2 + ... is its isotpyic decomposition.
	Then by Schur (version 1)
		End_R(M) = \product End_R(I_j)
	Now suppose that M is a semisimple module over an k-algebra	
	with k being an algebraically closed field.   Further suppose that
	the I_j is a (finite) direct sum n_j times of the j^th simple module,
	and that each simple module S_j is finite dimensional over k.	
	Then, by Schur again (version 2):
		End_R(I_j) =  M_(n_j)(k)