Tavakoli, Javad A characterization of geometric fields in a topos. (English) Zbl 0558.18002 Commun. Algebra 13, 101-111 (1985). 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 Cited in 1 Document MSC: 18B25 Topoi Keywords:geometric field in topos; category with finite products; invertible elements; elementary topos; ideals; subobject classifier PDFBibTeX XMLCite \textit{J. Tavakoli}, Commun. Algebra 13, 101--111 (1985; Zbl 0558.18002) Full Text: DOI 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) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.