Abstract
Here are some applications of the theory of the previous chapter. In the first section, it is proven that, for every Baer–Stone half-shell that is two-sided and unital, there is a reduced and factor-transparent sheaf over a Boolean space that represents the half-shell and and has stalks with no divisors of zero. With all that has been done in previous chapters, the proof is relatively short. Just as the results of the previous chapters may be cast into categories, so we restate this result as the equivalence of two categories. In the second section are two more applications. Each von Neumann regular, commutative and unital ring is isomorphic to the ring of all global sections of a sheaf of fields over a Boolean space. And every biregular ring is isomorphic to the ring of all global sections of a sheaf with simple stalks over a Boolean space. These results extend to half-shells.
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- 1.
Kaplansky’s [Kapl68, p. 3] original definition of a Baer ring differs from that of Kist’s and Hofmann’s [Hofm72, pp. 327–8], and is both more and less than that of being Baer–Stone in our sense. Kaplansky requires that the annihilator of each subset of R be a principal ideal generated on one side by an idempotent, which nevertheless need not be central. In a short personal history near the beginning of his book on rings of operators, Kaplansky [Kapl68] writes that this is in honor of Reinhold Baer’s [Baer52] early use of the condition.
- 2.
This term arose when Grätzer and Schmidt [GrätSc57] answered a question of Stone.
- 3.
An ideal P of a ring is prime if, whenever JK ⊂ P for some ideals J and K, then J ⊂ P or K ⊂ P. Here JK is complex multiplication.
- 4.
This concept differs from the regular ideals and congruences studied previously.
- 5.
See the comments at the beginning of Sect. XII.5 for related notions and references.
- 6.
See [AreKa48] and [Ledb77]. See also [ForMc46] and [Goode79, p. 35].
- 7.
However, when the shell is a lattice, this is nothing new. For, as Mai Gehrke pointed out, any biregular bounded lattice A is a Boolean lattice. This is true since [a] = { x | x ≤ a} in a lattice. Thus [a] = [e] implies a = e.
- 8.
Consult the book [Diers86] for a strongly categorical approach to sheaves with simple stalks over Boolean spaces.
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© 2012 Springer Science+Business Media, LLC
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Knoebel, A. (2012). Baer–Stone Shells. In: Sheaves of Algebras over Boolean Spaces. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4642-4_8
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DOI: https://doi.org/10.1007/978-0-8176-4642-4_8
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