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Rings

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Which of the following are rings with respect to the usual definition of addition and multiplication? (see attached)

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Solution Summary

This provides an example of determining what are rings.

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If R and S are both commutative rings with unity, then so is R (+) S (here (+) denotes the direct sum).

(1a) the set of positive integers is not a ring, since it is not an (abelian) group under addition (it does not contain
inverses).

(b) the set 3Z consisting of all the integer multiples of 3 is a ring: it is an abelian group under ...

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