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I was reading a research paper titled Purity and Reid's Theorem by A.Blass and J.Irwin and i have the problem with the explanation of the proof of the first theorem, that is theorem 1.1. In the proof of the first theorem, G is a torsion-free abelian group of infinite rank $\kappa$. At the end of the proof of theorem 1.1, the author says that :
Equivalently, $e(\alpha)$ should not be in the affine subspace of $\bar{G}$ spanned by $(f_{\beta}-g_{\alpha}, \beta < \alpha ) \cup (g_{\alpha}) $where $\bar{G}$ is the divisible hull of $G$.
Question:
What does it mean by affine subspace of $\bar{G}$ spanned by $(f_{\beta}-g_{\alpha}, \beta < \alpha ) \cup (g_{\alpha})$?

The research paper can be obtained from the web.

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Classically, an affine space is a torsor for a vector space; that is, it's a set equipped with a free transitive action of a vector space. Intuitively, it's a vector space that doesn't know where its origin is. For example, any translation of a subspace of a vector space is an affine space.

Affine spaces don't have a notion of addition of vectors, but they do have a well-defined notion of affine combination of points, which you can intuitively think of as a weighted average. The affine subspace spanned by a collection of points in an affine space is the affine space of affine linear combinations of those points.

In this case I guess "affine space" means a torsor for an abelian group. The definitions above generalize readily to modules over a commutative ring.