Hate to be pedantic, but ℤ︎ is a set (not a group) and even with 0 adjoined it only forms a group under the operators of addition, subtraction, and multiplication.. If you want a set that also forms a group under division you need to ‘upgrade’ to the set ℚ︎.

ℤ︎ (…, -2, -1, 0, 1, 2, …) and + forms a group. It is also a ring with * and +. ℚ︎ with * and + forms a field.

Yes, that's what qubex said, in correcting the error in your previous comment.

ℤ︎ with * is not a group, there is no inverse for 0. When being pedantic it's good to be correct.

You are perfectly correct, I was wrong. Upon closer reflection it appears impossible to have a nontrivial set (meaning other than {0}) that forms a group both under addition and under multiplication because the identity element of the former does not have an inverse in the latter. However, take this with a grain of salt, because I haven't had time to prove this to my own satisfaction yet (and I am loathe to go googling for answers in my own field of expertise).

My abstract algebra professor would be horrified by my misstatement, and I shall go to bring flowers as a sign of atonement and rememberance to his grave at the earliest available opportunity.