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Ring Identities

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For any x,y E R (i.e. Ring R) the following equalities hold.
a) 0.x=0
b) a(-b)=(-a)b=-(ab)
Prove either a or b. State any properties used in your proof.

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Problem:

For any x, y  R (i.e. ring R), the following equalities hold:
a) 0  x = 0
b) a(-b) = (-a)b = -(ab)

Prove either (a) or (b). State any properties used in your proof.

Solution:

We will prove the above equalities based on the axioms of a ring.

a) Any ring is a commutative group with respect to "addition" law, that is for any a  R, there exists (-a)  R so that:
...

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