Tensor Product of Galois Extensions of Rings

I was hoping that someone familiar with a bit of commutative algebra could help me see why this is true.

Suppose that are commutative rings with identity, such that is faithful as an -algebra (i.e., there is a monomorphism ). In this way, we may identify with and view .

Let and say (that is, is a finite group of ring automorphisms of such that elements of are fixed pointwise by these maps).

Define . We say that is a __normal__ extension of with group if (clearly we have the containment by definition of ).

We say that is a __Galois__ extension of if:

(i) and

(ii) satisfying

(where

is the Kronecker delta, and

represents the identity automorphism of

).

The theorem is that, if

is a Galois extension of

with Galois group

, then we may view

as a Galois extension of

(identifying

with

with Galois group

. The group

acts on the algebra

via the action

.

Right now, I'm just trying to show that

is a normal extension of

(that is, showing

). The containment

is clear, but because of how complicated expressions in

can be (as full summations of simple tensors), I can't see why the other containment must be true.

Anyone have any ideas at all? I could really use some help trying to understand this.