To prove: S 1 = { [ 0 ] , [ 2 ] } is a subring of ℤ 4 , and S 2 = { [ 0 ] , [ 3 ] } is a subring of ℤ 6

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Answer 1

Question

Chapter 5.1, Problem 54E

(a)

To determine

To prove: S1={[0],[2]} is a subring of ℤ4, and S2={[0],[3]} is a subring of ℤ6

(b)

To determine

The elements in S1⊕S2 and construct addition and multiplication tables for this ring.

(c)

To determine

Whether S1⊕S2 a commutative ring.

(d)

To determine

Unity element in S1⊕S2, if one exists.

(e)

To determine

All units in S1⊕S2, if they exist.

(f)

To determine

All zero divisors in S1⊕S2, if they exist.

(g)

To determine

All idempotent elements in S1⊕S2, if they exist.

(h)

To determine

To find: All nilpotent elements in S1⊕S2, if they exist.

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1. Given that h(t) = -5t + 3 t². A tangent line H to the function h(t) passes through the point (-7, B). a. Determine the value of ẞ. b. Derive an expression to represent the gradient of the tangent line H that is passing through the point (-7. B). c. Hence, derive the straight-line equation of the tangent line H 2. The function p(q) has factors of (q − 3) (2q + 5) (q) for the interval -3≤ q≤ 4. a. Derive an expression for the function p(q). b. Determine the stationary point(s) of the function p(q) c. Classify the stationary point(s) from part b. above. d. Identify the local maximum of the function p(q). e. Identify the global minimum for the function p(q). 3. Given that m(q) = -3e-24-169 +9 (-39-7)(-In (30-755 a. State all the possible rules that should be used to differentiate the function m(q). Next to the rule that has been stated, write the expression(s) of the function m(q) for which that rule will be applied. b. Determine the derivative of m(q)

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Answer 2

Question

Chapter 5.1, Problem 54E

(a)

To determine

To prove: S1={[0],[2]} is a subring of ℤ4, and S2={[0],[3]} is a subring of ℤ6

(b)

To determine

The elements in S1⊕S2 and construct addition and multiplication tables for this ring.

(c)

To determine

Whether S1⊕S2 a commutative ring.

(d)

To determine

Unity element in S1⊕S2, if one exists.

(e)

To determine

All units in S1⊕S2, if they exist.

(f)

To determine

All zero divisors in S1⊕S2, if they exist.

(g)

To determine

All idempotent elements in S1⊕S2, if they exist.

(h)

To determine

To find: All nilpotent elements in S1⊕S2, if they exist.

Expert Solution & Answer

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Answer 3

Question

Chapter 5.1, Problem 54E

(a)

To determine

To prove: S1={[0],[2]} is a subring of ℤ4, and S2={[0],[3]} is a subring of ℤ6

(b)

To determine

The elements in S1⊕S2 and construct addition and multiplication tables for this ring.

(c)

To determine

Whether S1⊕S2 a commutative ring.

(d)

To determine

Unity element in S1⊕S2, if one exists.

(e)

To determine

All units in S1⊕S2, if they exist.

(f)

To determine

All zero divisors in S1⊕S2, if they exist.

(g)

To determine

All idempotent elements in S1⊕S2, if they exist.

(h)

To determine

To find: All nilpotent elements in S1⊕S2, if they exist.

Expert Solution & Answer

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Answer 4

Question

Chapter 5.1, Problem 54E

(a)

To determine

To prove: S1={[0],[2]} is a subring of ℤ4, and S2={[0],[3]} is a subring of ℤ6

(b)

To determine

The elements in S1⊕S2 and construct addition and multiplication tables for this ring.

(c)

To determine

Whether S1⊕S2 a commutative ring.

(d)

To determine

Unity element in S1⊕S2, if one exists.

(e)

To determine

All units in S1⊕S2, if they exist.

(f)

To determine

All zero divisors in S1⊕S2, if they exist.

(g)

To determine

All idempotent elements in S1⊕S2, if they exist.

(h)

To determine

To find: All nilpotent elements in S1⊕S2, if they exist.

Expert Solution & Answer

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Answer 5

Chapter 5.1, Problem 54E

(a)

To determine

To prove: S1={[0],[2]} is a subring of ℤ4, and S2={[0],[3]} is a subring of ℤ6

(b)

To determine

The elements in S1⊕S2 and construct addition and multiplication tables for this ring.

(c)

To determine

Whether S1⊕S2 a commutative ring.

(d)

To determine

Unity element in S1⊕S2, if one exists.

(e)

To determine

All units in S1⊕S2, if they exist.

(f)

To determine

All zero divisors in S1⊕S2, if they exist.

(g)

To determine

All idempotent elements in S1⊕S2, if they exist.

(h)

To determine

To find: All nilpotent elements in S1⊕S2, if they exist.

Answer 6

Chapter 5.1, Problem 54E

(a)

To determine

To prove: S1={[0],[2]} is a subring of ℤ4, and S2={[0],[3]} is a subring of ℤ6

Answer 7

Chapter 5.1, Problem 54E

(a)

To determine

To prove: S1={[0],[2]} is a subring of ℤ4, and S2={[0],[3]} is a subring of ℤ6

Answer 8

(b)

To determine

The elements in S1⊕S2 and construct addition and multiplication tables for this ring.

Answer 9

(b)

To determine

The elements in S1⊕S2 and construct addition and multiplication tables for this ring.

Answer 10

(c)

To determine

Whether S1⊕S2 a commutative ring.

Answer 11

(c)

To determine

Whether S1⊕S2 a commutative ring.

Answer 12

(d)

To determine

Unity element in S1⊕S2, if one exists.

Answer 13

(d)

To determine

Unity element in S1⊕S2, if one exists.

Answer 14

(e)

To determine

All units in S1⊕S2, if they exist.

Answer 15

(e)

To determine

All units in S1⊕S2, if they exist.

Answer 16

(f)

To determine

All zero divisors in S1⊕S2, if they exist.

Answer 17

(f)

To determine

All zero divisors in S1⊕S2, if they exist.

Answer 18

(g)

To determine

All idempotent elements in S1⊕S2, if they exist.

Answer 19

(g)

To determine

All idempotent elements in S1⊕S2, if they exist.

Answer 20

(h)

To determine

To find: All nilpotent elements in S1⊕S2, if they exist.

Answer 21

(h)

To determine

To find: All nilpotent elements in S1⊕S2, if they exist.

Answer 22

Expert Solution & Answer

Answer 23

Expert Solution & Answer

Answer 24

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Answer 25

To prove: S 1 = { [ 0 ] , [ 2 ] } is a subring of ℤ 4 , and S 2 = { [ 0 ] , [ 3 ] } is a subring of ℤ 6

Solution Summary: The author explains that S_1 is a non-empty set.

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To prove: S1={[0],[2]} is a subring of ℤ4, and S2={[0],[3]} is a subring of ℤ6

The elements in S1⊕S2 and construct addition and multiplication tables for this ring.

Whether S1⊕S2 a commutative ring.

Unity element in S1⊕S2, if one exists.

All units in S1⊕S2, if they exist.

All zero divisors in S1⊕S2, if they exist.

All idempotent elements in S1⊕S2, if they exist.

To find: All nilpotent elements in S1⊕S2, if they exist.

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Chapter 5 Solutions