I thought that I understand the module concept but I've been confused since I had to deal with such task:
"Find an example of finitely generated module that is non-finitely generated abelian group."
Is that possible? thanks in advance for your help
I thought that I understand the module concept but I've been confused since I had to deal with such task:
"Find an example of finitely generated module that is non-finitely generated abelian group."
Is that possible? thanks in advance for your help
consider the module over the rational numbers (R = Q) generated by 1: that is the module Q itself (since Q being a ring, is a module over Q).
if Q was a finitely-generated module over Z (abelian groups are Z-modules), we would be able to express any rational number as a Z-linear combination of a finite set of rationals:
B = {q_{1},...,q_{n}}.
write q_{j} = a_{j}/b_{j}, where a_{j},b_{j} ≠ 0, and gcd(a_{j},b_{j}) = 1.
consider k = lcm(b_{1},...,b_{n}).
certainly any Z-linear combination of elements of B can be written as N/k, where N is some integer (which N we get depends on which Z-linear combination we have, but we can use the same k).
taking out common factors, we have N/k = N'/k', where gcd(N',k') = 1.
let p be a prime number that does not divide k (we can ALWAYS find such a prime number, since there are infinitely many primes, but k has only a finite number of prime factors).
i claim 1/p is not in span(B) (where this indicates the Z-linear span).
for if it were, we have 1/p = N'/k', and thus: k' = N'p, hence p divides k', and thus divides k, contradiction.