Re: Module basis

In article <1171904788.667996.178480@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
jseppa17@xxxxxxxxx <jseppa17@xxxxxxxxx> wrote:
If A is a given module and B, C are two basis of A, is card B = card C
in general?

Depends on what they are modules over.

For modules over commutative rings, the answer is "yes". For modules
over more general rings, the answer may be no.

The simplest example I know: Let V be an infinite dimensional vector
space over a field K, so that V is isomorphic to V(+)V (direct sum).

Let R = End_K(V) = {f:V->V | f is a K-linear map}.

Since Hom(A->B(+)C) is naturally isomorphic to Hom(A,B)(+)Hom(A,C),
and V(+)V is isomorphic to V, you have that

End_K(V) = Hom_K(V,V)
= Hom_K(V,V(+)V)
= Hom_K(V,V) (+) Hom_K(V,V)
= End_K(V) (+) End_K(V)

you have that R is isomorphic to R(+)R.

Trivially, {1} is a basis for R as a module over itself. Also,
{(1,0),(0,1)} is a basis for R(+)R as a module over itself. Since R is
isomorphic to R(+)R, you then have a basis with 1 element, and a basis
with 2 elements. (In fact, you can have bases with any finite number
of elements you please by repeating the process, since R is isomorphic
to R^n for every natural number n, as modules over R).

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes" by Bill Watterson)
======================================================================

Arturo Magidin
magidin-at-member-ams-org

.

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