Assignment:
Let G be the direct sum of a countably infinite number of copies of Z. Find an element of End_Z(G) which has a left inverse, but is not a unit.
Please explain in detail.
Think of elements of End_Z(G) as infinite matrices with integer entries.
Definition: Let G be an abelian group and let End_Z(G) be the set of all group homomorphisms from G to itself. We call it the endomorphism ring of G. Here, the ring operations are defined as follows. For f, g ∈ End_Z(G), we let f + g be the homomorphism given by (f + g)(x) = f(x) + g(x), where additive notation is used for the group operation in G. The product, f ° g, of f, g ∈ EndZ(G) is simply the composition f ° g.
Lemma: Under these operations, End_Z(G) is a ring, with additive identity the trivial homomorphism, and multiplicative identity the identity map.
Find an element of End_Z(G) which has a left inverse, but is not a unit.