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The inclusion is a semi-ring morphism. So, every -vector space becomes a -module by restriction of scalars, in particular . Now, the convex cones in are exactly the sub--modules of . And the polyhedral cones are exactly the convex cones which are finitely generated as a -module.
But note that I suppose that a convex cone is a subset of such that if and , so they are convex cones with .
You would also have the same if you replace by , but in this case you should note require a zero in the definition of semi-ring and module over a semi-ring.
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