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Elements in an Abelian group

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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.

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Solution Preview

The solution is attached.

We can think of G as a vector which has infinitely many components, each of which is an integer. Imagine taking a (row) vector and shifting all the elements1 position to the right to form a new vector, and filling in a 0 in the first position. ...

Solution Summary

The solution provides an example of finding an element with a left inverse.

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