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The ring of polynomials. A non-empty set S is called a commutative ring if there are two operations, denoted by + and · such that for all a, b, c ∈ S, (i) a + b = b + a ∈ S, (ii) (a + b) + c = a+ (b + c), (iii) there is a 0 in S such that a + 0 = a for all a in S, (iv) for each a in S there is an element −a in S such that a + (−a) = 0. Further, (v) a · b = b · a ∈ S, (vi) a · (b · c) = (a · b) · c, and (vii) a · (b + c) = a · b + a · c. A set satisfying (i)-(iv) is said to be a commutative group under the operation +. For example, the sets J, G, and H are commutative rings, as are the set of all rational numbers, which we denote by R, and the set of all complex numbers, which we denote by C. Note that the last two examples have a further property: they admit division by non-zero elements.
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