Abstract Algebra - Rings are an important algebraic structure, and modular arithmetic

[frogyspond]

Thank you

 

[stapel]

I included my original work with my critiques remarks

Are you saying that the lines following each exercise statement are your work, and the red bits are the comments from the grader? So it's as follows?

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

My answer: Definition of Ring

Commutative law for addition a, b are real numbers a+b=b+a
Associative law for addition a,b,c are real numbers (a+b)+c = a+(b+c)
Zero Law a is a real number a+0 = 0
Negative Law a is a real number a+a = 0
This element is called negative of a, denoted as b = -a
Associative law for multiplication a,b,c are real numbers (a*b)*c = a*(b*c)
Distributive Law a,b,c are real numbers (a+b)*c = ac + bc
or
a,b,c are real numbers c(a+b) = ca + cb

Grader's comments: Clear statements of most of the properties of a ring. The closure properties have not been explicitly stated. Some steps are shown for how addition and multiplication in modular arithmetic is handled. Detailed work to show that each of the properties must hold for all of the elements in Z7 are not evident.

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2. Provide written justification for each step of your proof.
[a]₇ + [b]₇ =

My answer:
[a]₇ + [b]₇ - use the distributive law
= [a + b]₇

[a]₇ × [b]₇ =

My answer:
[a]₇ × [b]₇ – use the distributive law
= [a × b]₇

Grader's comments: Some steps to show how addition and multiplication in modular arithmetic is handled. The justification of the distributive law does not apply in this step. Clear work to show that each of the properties of a ring hold in modular arithmetic is needed, and each of these steps must be fully justified.

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B. Use the definition for an integral domain to prove that Z₇ is an integral domain.
Provide written justification for each step of your proof.

Use the definition for a ring to prove that Z7 is a ring under the operations + and x defined as follows:
[a]₇ + ₇ = [a + b]₇
and
[a]₇ x ₇ = [a x b]₇

My answer:

• [a]₇ *₇ = [0]₇ in〖 Z〗₇
  [*][ab] ₇ = [0] ₇
  [*]ab ∈ [0] ₇
  [*]ab is a multiple of 7
  [*]7|ab so 7|a or 7| b
  [*]a ∈ [0] ₇ or b ∈ [0] ₇.
  [*][a] ₇ = [0] ₇ or ₇= [0] ₇.
  [*]〖 Z〗₇ has no zero divisors and is an integral domain.
  

Grader's comments: Some of the properties of an integral domain are listed. Clear work to show that Z7 is a commutative ring with unity and no zero divisors is not evident. Please show complete and explicit work to verify that Z7 is a commutative ring, that there are no zero divisors, and that there is unity.

My answer:

• a and b in Z₇, a*b = 0 only if a or b is 0.
• a and b cannot be equal to 0 in Z₇.
• 7 is prime, meaning a nor b is divide 7
• Since 7 is prime, this means that a*b does not divide 7.
• Thus, a*b is not equal to 0 in Z₇.

Grader's comments: Includes support for the fact that Z7 has no zero divisors. All of the steps in the work to show that Z7 is a commutative ring with unity will require explicit justification.

And are you asking for our responses to the grader's comments? Thank you. ;-)