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Show that the set of all integers that are multiplies of 3 is a commutative ring....

Show that the set of all integers that are multiplies of 3 is a commutative ring. Does it have an identity?

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Answer #1

Proof:

Let .

To show that : is a commutative ring.

Let   .

Then and   , where   .

.

, where .

is closed under addition and multiplication.

Associativity and commutativity of additon and multiplication , and distributivity all hold in and hence hold in the subset .

Also, .

If   then   .

Hence, is a commutative ring.

However there is no multiplicative identity:

If   is the multiplicative identity then   for all .

.

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