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A note on the algebraic De Morgan’s law. (English) Zbl 0575.13002

A Boolean algebra \(A\) is complete iff its Stone space is extremely disconnected (e.d.). In this paper the authors consider various relatives of this result. In particular they show the following nice characterizations:
Theorem 1. The following are equivalent for a commutative ring \(R\): (a) \(R\) is a Baer ring; (b) \(Ann(AB)=Ann(A)+Ann(B)\) for all ideals \(A\) and \(B\) of \(R\); (c) \(R\) has no nilpotents and \(Ann(A\cap B)=Ann(A)+Ann(B)\); (d) \(Ann(A)\oplus Ann(Ann(A))=R\); (e) \(Ann(A)\) is a principal ideal and \(Ann(a)+Ann(b)=Ann(ab)\).
From this it follows Theorem 2: If \(R\) has no nilpotents, then \(spec(R)\) is extremely disconnected iff \(Ann(A\cap B)=Ann(A)+Ann(B)\),
and Theorem 3: If a topological space \(X\) is extremely disconnected then \(C(X)\) is a Baer ring. Conversely, if \(C(X)\) is a Baer ring and if \(X\) is completely regular, then \(X\) is extremely disconnected.

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
54C40 Algebraic properties of function spaces in general topology
54G05 Extremally disconnected spaces, \(F\)-spaces, etc.
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