Localization in Algebra

Abstract

This chapter provides a digression concerning the all-important technique of localization in algebra. Localization generally means formally inverting a multiplicatively closed subset in a ring. However, the chapter focuses on the particular case of inverting all nonnegative powers of a variable u in an ℝ[u]-module. Localization of an ℝ[u]-module with respect to a variable u kills the torsion elements and preserves exactness. The chapter then looks at the proposition that localization preserves the direct sum. The simplest proof for this proposition is probably one that uses the universal mapping property of the direct sum. The chapter also considers antiderivations under localization.