## 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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### References:

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