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Usually, the identity of addition property is defined to be an axiom (which only specifies the existence of zero, not uniqueness), and the zero property of multiplication is a consequence of existence of zero, existence of an additive inverse, distributivity of multiplication over addition and associativity of addition.

Proof of 0 * a = 0:

0 * a = (0 + 0) * a [additive identity]

0 * a = 0 * a + 0 * a [distributivity of multiplication over addition]

0 * a + (-(0 * a)) = (0 * a + 0 * a) + (-(0 * a)) [existence of additive inverse]

0 = (0 * a + 0 * a) + (-(0 * a)) [property of additive inverses]

0 = 0 * a + (0 * a + (-(0 * a))) [associativity of addition]

0 = 0 * a + 0 [property of additive inverses]

0 = 0 * a [additive identity]

A similar proof works for a * 0 = 0 (with the other distributive law if commutativity of multiplication is not assumed).

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Q: What is the difference between the zero property of multiplication and the identity property of addition?
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