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

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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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Robert · Accepted Answer

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),

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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