Tuesday, 3 September 2019

Abelian group in module

How to say every abelian group G is a module over the ring of integers I?

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Let G be an abelian group, the operation in G being denoted by+ and the identity element of G by 0. For any integer n and for any element an of G  we define na in a several ways:
If n is a positive integer, we define na=a+a+.....nterms.if n=0, we define 0a=0 where 0 on the right-hand side is the identity of G.
If n is a negative number, we can say n=-m where m is +ve integer, we define  (-m)a=-(ma) where -ma denoted  the inverse of ma in G.It can be easily seen that -(ma)=m(-a).

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