QUESTION 8 Let G is an abelian group with the additive operation. Define the operation of multiplication by the rule ab...

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QUESTION 8 Let G is an abelian group with the additive operation. Define the operation of multiplication by the rule ab- a- b

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Answer 1

Answer #1

Answer:
G is not a ring

Explanation:

The operation of multiplication as defined by the rule:

$ab=a-b$

for all a, b $\in$ G

is not associate.

$a(be) = a(b-c) = a _ (b-c) = a-b-c$

because:

$bc=b-c$ by the definition of operation of multiplication

and

$a(b _ c) = a-(b-c) Cl$ by the definition of operation of multiplication

$(ab)c = (a-b)c = a-b-c$

because

$ab=a-b$ by the definition of operation of multiplication

and

$(a - b)c-a - b - c$ by the definition of operation of multiplication

So,

$a(bc) (ab)c$

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Answer 2

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