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definition of reduced modules

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definition of reduced modules

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!

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

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

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.

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

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