definition of reduced modules

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# definition of reduced modules

Submitted by nona_nonomo on Wed, 09/28/2011 - 12:30

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i want to know what is definition of a reduced module over integral domain (because definition of torsion-free modules over general ring and definition of them in integral domain are different). Now, i have some definitions: 1) a module is called reduced iff i have only injective modules 0. 2) a module C is called reduced iff Hom(A,C)=0 for all divisible module C.

I do not know which definition for module over integral domain or none of them? are they equivalent iff R is integral domain?

Thank you very much, good luck to you!

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## Versions

(v1) by nona_nonomo 2011-09-28

## Re: definition of reduced modules

Your idea is so helpfull for me! Thank you very much. Let me research.

## Re: definition of reduced modules

Thanks Nona, and you always will be very welcome in your community.

perucho

## Re: definition of reduced modules

Hi nona,

basically a reduced module is a generalization of a reduced ring; the essential difference is that a ring is defined over a scalar field, whereas in a module, in general, the scalars are defined over a ring which is not necessarily commutative, i.e. a module doesn't need to be defined over a field, just over a ring.

A reduced ring R has no nonzero nilpotent elements, that is, r^2 = 0, implies r = 0, for all r in R. Typically a module is a kind generalization of a vector space. As an additional example, modules generalize abelian(additive)groups, being modules over the ring of integers.

Now, you are dealing with integral domains which are *commutative* rings without zero divisors and with the absence of the trivial ring {0} (a prime ideal), therefore your reduced modules, in this case, will be double sided modules. Also, in integral domains, Ideals and quotient rings are modules.

All of this is a basic explanation; I think the following paper may helps you a lot better.

http://www.ieja.net/papers/2008/V3/3-V3-2008.pdf

I also hope anyone of our PM mathematicians specialized in this matter may extends the subject in order to help you a bit more.

perucho