On the prime spectrum of the ring of bounded nonstandard complex numbers

Echi, Othman; Khalfallah, Adel

doi:10.1090/proc/14204

On the prime spectrum of the ring of bounded nonstandard complex numbers
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by Othman Echi and Adel Khalfallah PDF

Proc. Amer. Math. Soc. 147 (2019), 687-699 Request permission

Abstract:

In this paper, we provide some algebraic structures of convex subrings of ${}^*\mathbb {C}$, a nonstandard extension of the field of complex numbers $\mathbb {C}$. In particular, a detailed description of the prime spectrum of any convex subring of ${}^*\mathbb {C}$ is given.

To achieve our goal, first we investigate prime ideals and we characterize two consecutive elements in the spectrum of a divided domain.

We also show that the prime spectrum of the ring of bounded hypercomplex numbers has two peculiar properties: there are no three consecutive elements in the spectrum; moreover, nonzero elements are a disjoint union of three subsets where one of them is strongly dense and the other two are dense in the spectrum.

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