General Ring Theory/Ring extensions

commutative ring extension is a module Let be a commutative ring and let be a commutative extension ring of . Then with its own addition and the restriction of the multiplication of to is a module over . From the axioms holding for rings, we deduce the module axioms as follows as follows: Let and . Then (distributivity) (distributivity) (commutativity of multiplication) (unit), the ring axiom that's being used being indicated in the brackets. Let be a ring extension. Then the function defines a function from ideals of to ideals of . Indeed, because it is certainly closed under addition and multiplication by elements of . Let be a ring extension, and suppose that is a multiplicative set. Then is a multiplicative set of . is closed under multiplication because both and are.