Is (-x) * y = x * (-y) true for all rings? It seems simple enough but I feel like * must be commutative when trying to prove this.
Not by definition. 1 + (-1) = 0 since 1 and -1 are additive inverses of each other -1(1 + (-1)) = -1(0) = 0, since 0 times anything is 0. -1(1) + (-1)(-1) = 0 Since -1(1) and (-1)(-1) add to zero, they are additive inverses. We know that -1(1) = -1, since 1 is the multiplicative identity, so -1(-1) must equal 1.