Now we must show that this abelian group is a ring. The next step is to identify if an abelian group is a ring. In order to use multiplication as our binary operation, Z must also be closed under multiplication. With these two operations we need to show that the following three properties hold in order to prove that an abelian group is a ring. The netural element of this group is 0.
Multiplication is associative.
Now we need to explain what a subset and an Ideal are. The following two conditions must hold true in order for the ring to be an ideal. This is equivalent to the fact that I is a subring of R. For an example to see if 2Z is a subring of R we could go back through all the steps in the definition of a ring , but it is a lot easier to show that above condition. In this case if you take two integers and subtract them you will still get an even integer. Now that we have found out what an ideal is in a ring we need to look to see what an ACC of Ideals is.