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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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    The dual of the category of abelian groups can be embedded in the category of locally compact abelian groups, by Pontryagin duality.2012-10-18

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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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    *(cont.)* according o what you wrote, one of those statements is true and the other isn't!2012-10-22