Could you explain how to show 9.16 in detail? I included list of theorems and definitions from the textbook. Definition 9.3. Let R be a ring and I an ideal of R. Then the factor ring (or quotientring), R/I, is the set of all left cosets {a + I : a e R} together with the operations(a + I) + (b +I) = a +b+ I and (a + I)(b+ I) = ab + I, for all a, b e R.Theorem 9.6. For any ring R and ideal I, the factor ring R/I is a ring.Example 9.9. Let R = Z and I = (5) = 5Z. Then R/I = {0+ I, 1 + I, 2 +I,3 + 1,4 + I} and, for instance, (2 + I) + (4 + I)(3 + I)(4+ I) = 12 + I = 2 +I.= 6+ I = 1 + I and%3DExample 9.10. Let R = M2(Z) and let I be the ideal consisting of all matriceswhose entries are even. Then notice that for any a;; e Z, we haveaji a12b11 b12+1 =+1,a21 a22b21 b22,where bij is 0 if a¡; is even and 1 if a;j is odd. Thus, R/I consists of the sixteendifferent elements (bii b12b21 b22+1,b¡¡ € {0, 1}. We perform arithmetic in the followingfashion:((49) +*) * (6 ) + +) – (; 2) +- - (")-(6:) + ') = (; )+ I1 0+ 1) ++1:and(::) -·) (6) +-) - (; ) + -(;) --)(:) --) - (:)+ I.1 2Example 9.11. Let R = R[x] and I = (x² + 3). Readers familiar with polynomiallong division will know that if f (x) e R, then f (x) = (x² + 3)q(x) +r(x), whereq and r are polynomials, with r(x)unfamiliar with polynomial long division can peek ahead to Section 10.1, where itwill be discussed in more generality.) Since (x² + 3)q(x) e I by absorption, weknow that elements of R/I are of the form a + bx + I, with a, b e R. Additionbehaves as expected; for instance, (2+3x + I)+(7 – 4x + I) = 9 – x+ I. To dealwith multiplication, observe that x2 – (-3) e I; thus, x² + I = -3+I. Therefore,we have calculations such as= a + bx, for some a, b E R. (Those who are(5 + 4x + I)(-7+2x + I) = -35 – 18x + 8x² + I= -35 – 18x + 8(-3) + I= -59 – 18x + I.Let us also record a few basic facts about factor rings.Theorem 9.7. Let R be a ring and I an ideal. Then1. if R is commutative, then so is R/I;2. if R has an identity, then so does R/I; and3. if u is a unit of R, then u + I is a unit of R/I.Theorem 9.8. Let R be a ring with ideals I and J, such that I C J. Then J/I isan ideal of R/I. 9.16. Let R be a ring and Iaproperideal.1. If R is an integral domain, does it follow that R/I is an integral domain? Provethat it does, or find a counterexample.2. If R/I is an integral domain, does it follow that R is an integral domain? Provethat it does, or find a counterexample.

Question

Could you explain how to show 9.16 in detail? I included list of theorems and definitions from the textbook. Definition 9.3. Let R be a ring and I an ideal of R. Then the factor ring (or quotientring), R/I, is the set of all left cosets {a + I : a e R} together with the operations(a + I) + (b +I) = a +b+ I and (a + I)(b+ I) = ab + I, for all a, b e R.Theorem 9.6. For any ring R and ideal I, the factor ring R/I is a ring.Example 9.9. Let R = Z and I = (5) = 5Z. Then R/I = {0+ I, 1 + I, 2 +I,3 + 1,4 + I} and, for instance, (2 + I) + (4 + I)(3 + I)(4+ I) = 12 + I = 2 +I.= 6+ I = 1 + I and%3DExample 9.10. Let R = M2(Z) and let I be the ideal consisting of all matriceswhose entries are even. Then notice that for any a;; e Z, we haveaji a12b11 b12+1 =+1,a21 a22b21 b22,where bij is 0 if a¡; is even and 1 if a;j is odd. Thus, R/I consists of the sixteendifferent elements (bii b12b21 b22+1,b¡¡ € {0, 1}. We perform arithmetic in the followingfashion:((49) +*) * (6 ) + +) – (; 2) +- - (&#34;)-(6:) + ') = (; )+ I1 0+ 1) ++1:and(::) -·) (6) +-) - (; ) + -(;) --)(:) --) - (:)+ I.1 2Example 9.11. Let R = R[x] and I = (x² + 3). Readers familiar with polynomiallong division will know that if f (x) e R, then f (x) = (x² + 3)q(x) +r(x), whereq and r are polynomials, with r(x)unfamiliar with polynomial long division can peek ahead to Section 10.1, where itwill be discussed in more generality.) Since (x² + 3)q(x) e I by absorption, weknow that elements of R/I are of the form a + bx + I, with a, b e R. Additionbehaves as expected; for instance, (2+3x + I)+(7 – 4x + I) = 9 – x+ I. To dealwith multiplication, observe that x2 – (-3) e I; thus, x² + I = -3+I. Therefore,we have calculations such as= a + bx, for some a, b E R. (Those who are(5 + 4x + I)(-7+2x + I) = -35 – 18x + 8x² + I= -35 – 18x + 8(-3) + I= -59 – 18x + I.Let us also record a few basic facts about factor rings.Theorem 9.7. Let R be a ring and I an ideal. Then1. if R is commutative, then so is R/I;2. if R has an identity, then so does R/I; and3. if u is a unit of R, then u + I is a unit of R/I.Theorem 9.8. Let R be a ring with ideals I and J, such that I C J. Then J/I isan ideal of R/I. 9.16. Let R be a ring and Iaproperideal.1. If R is an integral domain, does it follow that R/I is an integral domain? Provethat it does, or find a counterexample.2. If R/I is an integral domain, does it follow that R is an integral domain? Provethat it does, or find a counterexample.

Accepted Answer

Let R be a ring and I be an ideal.
1. Choose R=(ℤ, +, ·) and I=4ℤ.
Result: Ideals of ℤ are nℤ where…

9.16. Let R be a ring and I a proper ideal. 1. If R is an integral domain, does it follow that R/I is an integral domain? Prove that it does, or find a counterexample. 2. If R/I is an integral domain, does it follow that R is an integral domain? Prove that it does, or find a counterexample.