Minus times Minus is Plus in a Ring

For in a ring, is not an extra sign convention. It follows from distributivity and the uniqueness of additive inverses. Here means the element cancelling , not necessarily a negative number.

Multiplication preserves zero

Distributivity gives . Cancelling gives ; similarly, .

Each minus sign forces a cancellation

First, multiply by :

Thus cancels , so .

Next, multiply by and substitute:

Since already cancels , uniqueness forces

No division, multiplicative commutativity or multiplicative identity was needed; a field is only a special case. “Plus” means the original product, not positivity. The sign rule is a consequence of preserving cancellation.