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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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 (m,a)0(n,b)=(m•n,mobpa noama0b). Show that Z x R is a ring with unity with these operations. (You may assume that m obandnoacRwheneverm,nEZanda,bER.) 2) Show that the unity element in a commutative ring (R, +, o) is unique. 3) Give an example to show that the sum of two zero divisors need not be a zero divisor. 4) Show that kernel of any ring homomorphism 4): R —> S is an ideal. 5) Prove that an ideal containing a unit element is the whole ring. 6) Prove that every ideal of Zn is principal. Is Zn principal? 7) Prove that the ideal is prime in Z if and only if n = 0, ± 1, or Inl is prime. 8) Prove that every proper prime ideal of Z is maximal. 9) Let 3 denote the ideal of 44] of Gaussian integers a + bi such that a = b(mod 2). Describe the factor ring Z[i] / 3.



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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
126 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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