Here's an idle thought I had many years ago but never bothered to follow up on. Ideals in a ring R can be multiplied. They can also be added; I + J is the ideal generated by I and J. Multiplication of ideals distributes over addition. However, we don't have additive inverses. So it seems that the ideals of R form a semiring. Does studying the structure of this semiring tell us anything useful about the structure of R?
A casual search hasn't turned up the answer to my question, but maybe I'm not looking in the right places. Or maybe it's just not a useful idea. -- Tim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences