What Is the Value of Ext(R, Z) in Homological Algebra?

[playa007]

Homework Statement

Compute Ext(R,Z) where R is the real numbers and Z is the integers

Homework Equations

The Attempt at a Solution

This is part of a problem is given in Hilton-Stammbach page 109; the previous parts of the problem asks to compute Ext(Q,Z) and Hom(Q,Q/Z); I have computed both and they are both isomorphic to R

 

[mighty2000]

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To compute Ext(R,Z), we can use the fact that R is a flat Z-module, meaning that the functor Hom(R,-) is exact. This allows us to use the following isomorphism: Ext(R,Z) = Ext(R/Z,Z).

Since R/Z is a field, we know that Ext(R/Z,Z) = 0. Therefore, Ext(R,Z) = Ext(R/Z,Z) = 0.

In general, Ext(R,M) = 0 for any R-module M, since R is a flat module over itself. So, Ext(R,Z) = 0 in this case.