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Z-Modules and Modules Associated with Representations

1)I understand what a standard R-module (ring-module) is, but I have heard talk of modules associated with representations.
Could someone please give me some idea of what these are?

2) I am trying to find all modules over Z-the Integers; so far, I have only come up with additive groups.
How can I find all others?

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True. Why? we know that 1*a = a for a in the Z-module A.
So 2*a = (1+1)*a (in Z we know that 1+1=2...)
= 1*a + 1*a = a + a , using distributivity of * over +.
Similarly: (-1)*a + a = -1*a + 1*a = (-1+1)*a = 0*a = 0, so that
(-1)*a must be -a, where - is the inverse for +, of course.
So in fact, if we consider n*a, for n in Z, it is always clear
what this is, in any (!) module: the n-fold addition of a,
or its inverse for negative n.
So there is no choice at all for a Z-module: any additive group
can be made into a Z-module in this (canonical) way, and
any Z-module is obtained this way.

First define a double module:
let F be a commutative field and R a ring with ...

Solution Summary

Z-Modules and Modules Associated with Representations are investigated.