Good question. I think I screwed up a bit, and typed things up too fast. For multiplication the identity element is 1, and therefore inverses relate to 1 not 0

Good question. I think I screwed up a bit, and typed things up too fast. For multiplication the identity element is 1, and therefore inverses relate to 1 not 0 like addition. So what 'a' makes a*0 = 1? There is no such element and it has to be excluded as per other comments.