I was reading through Quaternions for Computer Graphics by Dr. John Vince and at page 10, he defined a ring as a set with a binary operation that forms an abelian group under addition and multiplication with additional distributive properties. Right after such definition, he stated that the set of integers form a ring. However, earlier, at page 9, he clearly stated that the integers DO NOT form a group under multiplication. Well then, how does the integers form an abelian group under multiplication when it does not even form a group under such binary operation?
Thank you in advance.
The usual definition of a ring with unit, requires that the multiplication is associative and have a neutral element (the unit element) and is distributive over the addition (that forms an abelian group). But the existence of an inverse and the commutativity of multiplication are not required.
The integers $\mathbb{Z}$ with the usual addition and multiplication are the classical example of a ring.
If any element of the ring has an inverse with respect to the multiplication we say that it is a division ring. An example is the ring of invertible matrices with $n\times n$ entries in a field under the usual addition and multiplications for matrices.