Peter said:
Thanks Deveno and Euge ... I appreciate your help ...
You write:
" ... ... to get you started in the other direction, show that if $M_R$ is an $S,R$-bimodule, that if we define:
$\eta(s) = s\cdot(-)$
this gives a ring homomorphism $\eta: S \to \text{End}(M_R)$. ... "
Can you explain what you mean by " ... define:
$\eta(s) = s\cdot(-)$ ... "
Peter
Define the image of $s \in S$ to be the mapping $\eta(s) \in \text{End}(M_R)$ that sends $m \mapsto s\cdot m$ (often called "scalar multiplication").
This process, of identifying a function of two variables (in this case $(-\cdot-): S \times M_R \to M_R$) with a function of a single variable (the $s$) that outputs ANOTHER function of a single variable (in this case $\eta(s)$, seen as a "left-action") is known as
currying, and is a common technique in mathematics:
underneath it all, we are invoking a set-isomorphism:
$M_R^{S \times M_R} \cong (M_R^{M_R})^S$
This let's us deal with an action (which involves TWO structures) "one structure at a time".
You should recognize the analogy here with a group action on a set, which is a similar kind of "product":
$G \times X \to X$.
We can view this as a kind of "scalar multiplication" where $X$ lacks any abelian group structure. We need "some" of the rules of modules to apply:
$g\cdot(g'\cdot x) = (gg')\cdot x$
$e_G\cdot x = x$
This is equivalent to saying we have a group-homomorphism:
$\sigma: G \to \text{Sym}(X)$ (the group of bijections on $X$; if $X$ is finite, these are called
permutations of $X$ These are precisely the set-automorphisms of $X$.).
The symmetric group $S_n$ has a natural action on any set with $n$ elements, since:
$S_n \cong \text{Sym}(\{1,2,\dots,n\})$ (so $S_n$ has a
representation as permutations which are precisely the elements of $S_n$ itself...in group-action theory, $S_n$ is an "invisible group", much like the standard basis for $\Bbb R^n$ is an "invisible basis").
Now, with an abelian group, not all endomorphisms are automorphisms. But we can form a similar notion of "monoid action" (which is actually a better analogy, since the multiplicative monoid of a ring with unity is typically NOT a group), which is a monoid-homomorphism:
$\phi: M \to X^X$ (the set $X^X$ forms what is called the
monoid of transformations of $X$, that is: ALL functions $f:X \to X$).
This is pretty basic: what monoids essentially are, are "things we can do right after another". What groups essentially are, are "things we can do that can be un-done". The process of "undoing" is "inversion".
For example, with an object, say $a$, we can "add more $a$'s, to get things like:
$aa$
$aaaaaa$
$aaa$
We can "partially" undo this (subtracting $a$'s), as long as we have enough $a$'s to start with. But if we have:
$aa$
it is not immediately clear how to "take away 3 $a$'s".
What I've done above should remind you of something:
free objects. Free monoids, for example, have their humble beginnings in something very basic: natural numbers. In a similar vein, free groups owe THEIR origins to something similar, integers.
I don't think I can stress enough, that these basic structures are essential for more abstract ones. If you have a doubt about modules, for example, one way to test your ideas is to see if they hold for square matrices with integer entries. While these form rather "special" modules, it is often easy to find simple counter-examples there. If you need a non-commutative ring $R$, these also make a good "test case" for $R$. A lot of sophisticated mathematics eventually boils down to some involved kind of arithmetic.