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A characterization of geometric fields in a topos. (English) Zbl 0558.18002

Part of the structure of a commutative ring k in a category E with finite products are the additive and multiplicative identities 0,1: \(e\to k\) where e is the terminal object of E. The subobject \(\gamma\) (k) of invertible elements of k can be constructed with equalizers in E. When E is an elementary topos there is also an object idl(k) of ideals of k. The result proved here is: \(0\neq 1\) and \(\gamma\) (k)\(\to k\leftarrow^{0}e\) is a coproduct diagram iff idl(k) is isomorphic to the subobject classifier of E. For E the category of sets, this is the easy result that a commutative ring is a field precisely when it has two ideals.
Reviewer: R.H.Street

MSC:

18B25 Topoi
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References:

[1] Johnstone P., L.M.S. Mathematical Monographs 10 (1977)
[2] Paré R., Indexed Categories and their Applications 661 pp 1– (1978) · doi:10.1007/BFb0061361
[3] Tavakoli J., Ph.D. Dissertation (1980)
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