Let (Z, +, •) denote the ring of integers under ordinary addition and multiplication. Define addition la) and multiplication 0 on the set Z x R by (m, a) ED (n, b) = (m + n, a NI b), and...

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Answered Same DayDec 20, 2021

Answer To: Let (Z, +, •) denote the ring of integers under ordinary addition and multiplication. Define...

Robert answered on Dec 20 2021
128 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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