7/11/2015 Evaluation Results Evaluation Results Author: Christa Graham Date Evaluated: 07/07/2015 04:23 PM (MDT) DRF template: Abstract Algebra (GR,...
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I need help fixing my work for the following question : Rings

are an important algebraic structure, and modular arithmetic has that structure. Recall that for the mod m relation, the congruence class of an integer x is denoted [x]m. For example, the elements of [–5]7 are of the form –5 plus integer multiples of 7, which would equate to {. . . –19, –12, –5, 2, 9, 16, . . .} or, more formally, {y: y = -5 + 7q for some integer q}. Task: A.Use the definition for a ring to prove that Z7 is a ring under the operations + and × defined as follows: [a]7 + [b]7 = [a + b]7 and [a]7 × [b]7 = [a × b]7 B. Use the definition for an integral domain to prove that Z7 is an integral domain. My submission and the comments from my grader are attached.

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Evaluation Results Author: Christa Graham Date Evaluated: 07/07/2015 04:36:23 PM (MDT) DRF template: Abstract Algebra (GR, QDT2-0914) Program: Abstract Algebra (GR, QDT2-0914) Evaluation Method: Using Rubric Evaluation Summary for Abstract Algebra: QDT Task 2 (0914) Final Score: Does not Meet Overall comments: 7/7/15 Necessary conditions of a ring are properly stated and two necessary conditions of an integral domain are also stated. More mathematical support is necessary to justify the conditions are met for Z7. Detailed Results (Rubric used: QDT Task 2 (0914)) Articulation of Response (clarity, organization, mechanics) (0) Unsatisfactory (1) Does Not Meet Standard (2) Minimally Competent (3) Competent (4) Highly Competent The candidate provides unsatisfactory articulation of response. The candidate provides weak articulation of response. The candidate provides limited articulation of response. The candidate provides adequate articulation of response. The candidate provides substantial articulation of response. Criterion Score: 4.00 A1. Steps of Ring Proof (0) Unsatisfactory (1) Does Not Meet Standard (2) Minimally Competent (3) Competent (4) Highly Competent The candidate does not state each step of the proof. The candidate states each step of the proof with no detail. The candidate states each step of the proof with limited detail. The candidate states each step of the proof with adequate detail. The candidate states each step of the proof with substantial detail. Criterion Score: 0.00 Comments on this criterion: 7/7/15 The conditions of a ring have been correctly stated yet steps of this proof are insufficient. A2. Justification of Ring Proof (0) Unsatisfactory (1) Does Not Meet Standard (2) Minimally Competent (3) Competent (4) Highly Competent The candidate does not provide appropriate, written justification for each step of the proof. The candidate provides appropriate, written justification, with no support, for each step of the proof. The candidate provides appropriate, written justification, with limited support, for each step of the proof. The candidate provides appropriate, written justification, with adequate support, for each step of the proof. The candidate provides appropriate, written justification, with substantial support, for each step of the proof. Criterion Score: 0.00
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Comments on this criterion: 7/7/15 The references to a theorem are not appropriate justifications for each condition of a ring. Revision is necessary to show that each element within Z7 meets the conditions of a ring. B1. Steps of Integral Domain Proof (0) Unsatisfactory (1) Does Not Meet Standard (2) Minimally Competent (3) Competent (4) Highly Competent The candidate does not state each step of the proof. The candidate states each step of the proof with no detail. The candidate states each step of the proof with limited detail. The candidate states each step of the proof with adequate detail. The candidate states each step of the proof with substantial detail. Criterion Score: 0.00 Comments on this criterion: 7/7/15 It is correctly stated that Z7 must be a commutative ring with no zero divisors. It is not evident that the third property is discussed. The no zero divisors proof contains necessary steps while the steps to show commutativity are insufficient. B2. Justification of Integral Domain Proof (0) Unsatisfactory (1) Does Not Meet Standard (2) Minimally Competent (3) Competent (4) Highly Competent The candidate does not provide appropriate, written justification for each step of the proof. The candidate provides appropriate, written justification, with no support, for each step of the proof. The candidate provides appropriate, written justification, with limited support, for each step of the proof. The candidate provides appropriate, written justification, with adequate support, for each step of the proof. The candidate provides appropriate, written justification, with substantial support, for each step of the proof. Criterion Score: 0.00 Comments on this criterion: 7/7/15 The statements of support that are provided for the no zero divisors are appropriate yet not all steps have been supported. Further support will also be necessary once complete steps are provided for the commutative and third property of this proof. C. Sources (0) Unsatisfactory (1) Does Not Meet Standard (2) Minimally Competent (3) Competent (4) Highly Competent When the candidate uses sources, the candidate does not provide in-text citations and references. When the candidate uses sources, the candidate provides only some in-text citations and references. When the candidate uses sources, the candidate provides appropriate in-text citations and references with major deviations from APA style. When the candidate uses sources, the candidate provides appropriate in-text citations and references with minor deviations from APA style. When the candidate uses sources, the candidate provides appropriate in-text citations and references with no readily detectable deviations from APA style, OR the candidate does not use sources. Criterion Score: 4.00
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Introduction: Rings are an important algebraic structure, and modular arithmetic has that structure. Recall that for the mod m relation, the congruence class of an integer x is denoted [ x ] m . For example, the elements of [–5] 7 are of the form –5 plus integer multiples of 7, which would equate to {. . . –19, –12, –5, 2, 9, 16, . . .} or, more formally, { y : y = -5 + 7 q for some integer q }. Task: A. Use the definition for a ring to prove that Z 7 is a ring under the operations + and × defined as follows: [ a ] 7 + [ b ] 7 = [ a + b ] 7 and [ a ] 7 × [ b ] 7 = [ a × b ] 7 According to Wolfram MathWorld (Weissen, E. 2015), A ring in the mathematical sense is a set together with two binary operators and (commonly interpreted as addition and multiplication, respectively) satisfying the following conditions: 1. Additive associativity: For all , , 2. Additive commutativity: For all , , 3. Additive identity : There exists an element such that for all , , 4. Additive inverse : For every there exists such that , 5. Left and right distributivity: For all , and , 6. Multiplicative associativity: For all , (a ring satisfying this property is sometimes explicitly termed an associative ring ).
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- Prove [ a ] 7 + [ b ] 7 = [ a + b ] 7 - Let [ a ] 7 = [ x ] 7 and Let [ b ] 7 = [ y ] 7 - Therefor a x mod 7 and b y mod 7 - By Theorem 2, part ii : If a b mod m and c d mod m, then a + c b + d mod m. (Nicodemi, O., Sutherland, M., Townsley, G. , 2007) - Therefor If a x mod 7 and b y mod 7, then a + b x+y mod 7 - Therefor = a b [ + ] 7 x [ + y ] 7 - Therefor [ a ] 7 + [ b ] 7 = [ a + b ] 7 - Prove [ a ] 7 × [ b ] 7 = [ a × b ] 7 - Let [ a ] 7 = [ x ] 7 and Let [ b ] 7 = [ y ] 7 - Therefor a x mod 7 and b y mod 7 - By Theorem 2, part iii : If a b mod m and c d mod m, then ac bd mod m.(Nicodemi, O., Sutherland, M., Townsley, G. , 2007) - If a x mod 7 and b y mod 7, then ab xy mod 7 - Therefor ab [ ] 7 = xy [ ] 7 - and therefor [ a ] 7 × [ b ] 7 = [ a × b ] 7 2. Provide written justification for each step of your proof. B. Use the definition for an integral domain to prove that Z 7 is an integral domain. - An integral domain is a ring that has no divisors and is commutative. - Prove that has no divisors: Z 7 - 7 is a prime number - Let in a [ ] 7 b [ ] 7 = 0 [ ] 7 Z 7 - By multiplication a [ ] 7 b [ ] 7 = ab [ ] 7 - Therefor ab [ ] 7 = 0 [ ] 7 - Therefor b and ab must be a multiple of 7 a 0 [ ] 7 - Therefor 7|ab so 7|a or 7|b - Therefor or a 0 [ ] 7 b 0 [ ] 7
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