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construction of product rings Z/nZ x Z/mZ

Hello,

is it possible in Sage to construct a product ring of two integer modulo rings?

When just creating an cartesian_product, it is for example not possible to add its elements:

R5=Integers(5)
R3=Integers(3)
CR=R5.cartesian_product(R3)
elem=CR((3,2)) # ok, element construction
elem*elem # ok
elem+elem # fails 
CR.list() #  not implemented
CR.list_of_elements_of_multiplicative_group # not implemented