I request your assistance on the following problem. I will provide everything I have so far, and indicate clearly those parts I'm having trouble with. First,to give you an idea of where I'm coming from, I'm currently enrolled in a Modern Algebra course using Hungerford as its text. The problem under discussion is number 23 on page 135.
An element in a ring is said to be idempotent if . An element of the center of the ring is said to be central. If is central idempotent in a ring with identity, then...
a) is central idempotent.
First, we must demonstrate for each . Thus, let and note:
Hence, is central.
Now it remains to demonstrate . Thus, note the following:
Therefore, is central idempotent.
b) and are ideals in such that .
Here is where I am having trouble. Is defined as or something else? I've searched through the chapter and I'm having trouble locating the definition, and I suspect it might mirror the definition of multiplication of ideals in some form or fashion, but I could be totally wrong.
Assuming I'm wrong, and , should I just take the usual approach of defining an isomorphism of rings? Or is there some special trick I'm missing?