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Sets and Rings

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Can any set that is not a group (Z for example) still be a ring or is it necessary that a set must be a group to be a ring?

Please give an example and counter example.

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A ring in the mathematical sense is a set S together with two binary operators + and * (commonly interpreted as addition and multiplication, respectively) satisfying the following conditions:
1. Additive Associativity: For all , a+(b+c)=(a+b)+c,
2. Additive Commutativity: For all , a+b=b+a,
3. Additive identity: There exists an element such that for all , ...

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