[graycat]

For your group (.,R), what is the inverse element of 0?

[doonesbury]

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.

[lalaithion]

Technically it should be either (.,R+) or even (.,R - {0}).