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Here is part of the problem - the Cartesian product business seems to be about categories, not the actual rings. And it doesn't really give you what you want. In fact, I find this all somewhat disturbing.
sage: type(CR)
sage.sets.cartesian_product.CartesianProduct_with_category
sage: type(R5)
sage.rings.finite_rings.integer_mod_ring.IntegerModRing_generic_with_category
sage: elem*elem
(9, 4)
sage: elem.summand_projection(1)
2
sage: type(elem.summand_projection(1))
sage.rings.integer.Integer
You can do it with just the groups, but of course that's not what you want.
