If A and B are ideals of some ring prove that AB A BSoluti

If A and B are ideals of some ring, prove that AB ? A ? B.

Solution

If x is an element of AB, then x = AB for some a in A and some B in B. Since A is an ideal, it absorbs elements from the right, so ab is in A. For the same reason, B absorbs elements from the left, so ab must be in B as well. Thus, ab = x is in in the intersection of A and B.

Then AB?A?B.