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