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What is the image of the restriction of the Spec functor (the functor from commutative rings to affine schemes) to commutative rings with the trivial monoid under multiplication?

Thanks very much

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Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. – Julian Kuelshammer Oct 18 '12 at 7:43
Hi Julian, thanks for the comment. I want to find a subcategory of the category of affine schemes that is dual to the category of commutative rings with trivial monoid under multiplication which is equivalent to the category of abelian groups. – u4953u Oct 18 '12 at 7:48
The dual of the category of abelian groups can be embedded in the category of locally compact abelian groups, by Pontryagin duality. – Zhen Lin Oct 18 '12 at 8:16

The question doesn't make sense. The spec construction works for unital commutative rings (otherwise many properties break down, e.g. the equivalence between affine schemes). And these don't have a trivial multiplication (except for the zero ring).

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The properties surely break down, but does the construction? – Mariano Suárez-Alvarez Oct 18 '12 at 8:07
What's a prime ideal in a non-unital ring? – Zhen Lin Oct 18 '12 at 8:10
One such the quotient is a domain? There are domains without unit :-) – Mariano Suárez-Alvarez Oct 18 '12 at 8:13
I really do not know if the construction makes sense (in particular, localizations of some form will be needed! (one should check the literature: it would not be surprising if people did come up with a sensible construction for localizations; universal localization might work, for example)) – Mariano Suárez-Alvarez Oct 18 '12 at 8:16
Localization at elements or prime ideals make no problems. In fact, they turn out to be unital. The structure sheaf can be written down as usual. However, the ring of global sections is unital and therefore cannot recover the given non-unital ring. Besides, the question was not about "how to develope algebraic geometry over non-unital rings" (which I had already asked on MO: – Martin Brandenburg Oct 18 '12 at 8:19

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