Answer To: Let (Z, +, •) denote the ring of integers under ordinary addition and multiplication. Define...
Robert answered on Dec 20 2021
1. Following we will try to prove the condition which are required for Z ×R
to be a Ring:
(a) Closed 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 ◦...