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During my studies of real analysis I've come across the definition of a field (or algebra) of sets.

My question is: What is the connection between this structure and the structure of a field or algebra known from abstract algebra?

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The two use the same word. – Mariano Suárez-Alvarez Nov 9 '10 at 16:10

The term "field" in "field / algebra of sets over A" has no relation to the term "field" as used in ring theory. Rather it denotes a Boolean subalgebra of the power set of A, i.e. a collection of subsets of A closed under union, intersection and complement. The term is well-known due to its frequent use in the famous Stone representation theorem: $\ $every Boolean algebra is isomorphic to a field of sets. For further information see this Wikipedia page and this sci.math thread.

Note: Boolean algebras can be considered as Boolean rings, i.e. rings where every element is idempotent, i.e. $\: x^2 = x\:$. However, except for the field of two elements, such rings are never fields: $\ 0 = x^2 - x = x\ (x-1)\ \Rightarrow\ x = 0\ \ {\rm or}\ \ x = 1\ $ in a field, i.e. fields have only trivial idempotents.

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