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A simple example of how this is true _even if you don't have negative numbers_:
Let's use mod 5 arithmetic. You have 5 elements in the ring -- 0,1,2,3,4
The additive inverses are as follows:
Which is to say that 1 is the additive inverse of 4 and 2 is the additive inverse of 3, and vice versa. 0 is the identity, of course.So what happens if you multiply 2 * -3 (2 times the additive inverse of 3).
The additive inverse of 3 is just 2, so the answer is 2 * -3 = 2 * 2 = 4.
The other way to calculate it is to find the additive inverse of the product:
2 * -3 = -(2 * 3) = -(1) which is the additive inverse of 1: 4 again.