• One of the first examples of a noncommutative division ring was found by the Irish mathematician William Hamilton in the 1840's. His example, the ring of real quaternions H(R) consists of all...

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Answered Same DayDec 20, 2021

Answer To: • One of the first examples of a noncommutative division ring was found by the Irish mathematician...

David answered on Dec 20 2021
133 Votes
1. Following we will try to prove the condition which are required for Z ×R
to be a Ring:
(a) Clos
ed under addition and Multiplication: As +, . is binary operation
for Z, and ./, ◦ is binary operation for R, Hence
(m+ n, a ./ b) ∈ Z×R
If we assume m ◦ b, n ◦ a ∈ R, then m ◦ b ./ n ◦ a ∈ R. Hence
m ◦ b ./ n ◦ a+ a ◦ b ∈ R. This gives
(m.n,m ◦ b ./ n ◦ a+ a ◦ b) ∈ Z×R
Hence operation ⊕ and � are binary operation in Z×R.
(b) Associative of addition and Multiplication: For m,n, p ∈ Z and
a, b, c ∈ R, we have: m + (n + p) = (m + n) + p and a ./ (b ./
c) = (a ./ b) ./ c, we have
(m, a)⊕ ((n, b)⊕ (p, c)) = ((m, a)⊕ (n, b))⊕ (p, c)
That is ⊕ is associative. Same way we have Now for �, we have
(m, a)� ((n, b)� (p, c)) = (m, a)� (n.p, n ◦ c ./ p ◦ b ./ b ◦ c)
= (m.(n.p),m ◦ (n ◦ c ./ p ◦ b ./ b ◦...
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