Modules Definition and 214 Threads

Let R be a commutative ring with 1. If F is a free module of rank n < 1, then show that HomR(F;M) is isomorphic to M^n, for each R-module M. I was thinking about defining a map Psi : HomR(F;M)--> M^n by psi(f) = (f(e1); f(e2); ... ; f(en)) where F is free on (e1; ... ; en) and show Psi is...

Homework Statement Given a field F, FS4 is a group algebra... we have a representation X that maps FS4 to 3x3 matrices over (presumably) F. Let V denote the FS4 module corresponding to X... do stuff. My question is, what the heck is V supposed to be? I assumed that V is F3, but that...

can anyone give me a precise statement of the "closure of modules" theorem in several complex variables? it says something like: a criterion for the germ of a function to belong to the stalk of an ideal at a point, is that the function can be uniformly approximated on neighborhoods of that point...

Hi all. Starting an undrgraduate programme in Theoretical Physics in two weeks and I have to choose two modules to take for the year. These modules can be of absolutely any topic that is availible in my college, they are called elective modules. I have come to a conclusion that it probably...

Please help me to prove that for a projective module P there exists a free module F, such that P +F =F. Here + denotes direct sum = denotes isomorphic. Thanks

Can somebody help me with the following proof: Let M be a semisimple module, say M = +_IS_i, where + denotes direct sum and S_i is a simple module. Then the number of summands is finite if and only of M is finitely generated. I have problem with understanding the proof of the following...

[SOLVED] Algebra - modules Homework Statement I'm reading this proof from D&F and there's something I don't get. It is theorem 1 of chapter 12 on modules over principal ideal domains. The theorem is the following "Let R be a ring and let M be a left R-module. Then, (Ever nonempty set of...

Homework Statement Does a torsion module M imply M is cyclic? Or does it imply M is finitely generated? I think cyclic implies torsion module. What about the reverse? The Attempt at a Solution I think there is a connection but don't see it.

I know there are many examples of rings like R[x], N, Q, where the elements can be fundalmentally different like polynomials and numbers. But are there different type of modules like there are rings? Or are modules just any vector? And the modules are different when you consider which R...

Anyone recommand readable intro books on modules and algebra of matrices?

I am supposed to show that the following are equivalent for a finitely generated module P: 1. P is Projective 2. P is isomorphic to direct summand of a free module (There are 2 others but they refer to a diagram) I am stuck on showing 1 => 2. I know that since P is projective there is...

In the spirit of challenges, here is a surprizingly simple question about finite modules, that I did not know the answer to until recently. It is so elementary that I suspect it was common knowledge to the ancients and only forgotten as algebra became more sophisticated. We all know that...

Consider the set of 2x2 matrices which form a ring under matrix multiplication and matrix addition. \mathbb{R}^3 is module defined over this ring. So, we have three dimensional vectors whose elements are 2x2 matrices. My question: Can I also define another "scalar multiplication" that...

I have a buddy asking if I can do this for his 300 ZX. I have checked and there is a programmable module available with software, but apparently some companies [one at least] are reprogramming the factory chip directly. Does anyone know where to get the code, or at least what values affect...