Choose a representative a' from the residue class a. Then

a · (0 + dR[X]) = (a' · 0) + dR[X]
according to the definition of multiplication. The multiplication in R yields a' · 0 = 0, so that we find

(a' · 0) + dR[X] = 0 + dR[X] = dR[X] = 0.
Hence a · 0 = 0.

Given a class a choose a representative a' in it. Now take b to be the class of -a'. Then the sum of a and b is the class of a' + -a', i.e., the class of 0.

The uniqueness of the opposite requires the other arithmetical rules. Here it is. Suppose the class b' is also an opposite of a. By associativity we have

(b + a) + b' = b + (a + b').

Since b is an opposite of a, the left-hand side evaluates to

(b + a) + b' = 0 + b' = b'.

Since b' is an opposite of a, the right-hand side evaluates to

b + (a + b') = b + 0 = b.

Conclusion: b = b'.