Given: A set R with two operations + and * is a ring if the following 8 properties are shown to be true:
1. closure property of +: For all s and t in R, s+t is also in R
2. closure property of *: For all s and t in R, s*t is also in R
Save your time - order a paper!
Get your paper written from scratch within the tight deadline. Our service is a reliable solution to all your troubles. Place an order on any task and we will take care of it. You won’t have to worry about the quality and deadlines
Order Paper Now3. additive identity property: There exists an element 0 in R such that s+0=s for all s in R
4. additive inverse property: For every s in R, there exists t in R, such that s+t=0
5. associative property of +: For every q, s, and t in R, q+(s+t)=(q+s)+t
6. associative property of *: For every q, s, and t in R, q*(s*t)=(q*s)*t
7. commutative property of +: For all s and t in R, s+t =t+s
8. left distributivity of * over +: For every q, s, and t in R, q*(s+t)=q*s+q*t
right distributivity of * over +: For every q, s, and t in R, (s+t)*q=s*q+t*q
The set of integers mod m is denoted Zm. The elements are written [x]m and are equivalence classes of integers that have the same integer remainder as x when divided by m. For example, the elements of [–5]m are of the form –5 plus integer multiples of 7, which would be the infinite set of integers {. . . –19, –12, –5, 2, 9, 16, . . .} or, more formally, {y: y = -5 + 7q for some integer q}.
Modular addition, [+], is defined in terms of integer addition by the rule
[a]m [+] [b]m = [a + b]m
Modular multiplication, [*], is defined in terms of integer multiplication by the rule
[a]m [*] [b]m = [a * b]m
Note: For ease of writing notation, follow the convention of using just plain + to represent both [+] and +. Be aware that one symbol can be used to represent two different operations (modular addition versus integer addition). The same principle applies for using * for both multiplications.
A. Prove that the set Z31(integers mod 31) under the operations [+] and [*] is a ring by using the definitions given above to prove the following are true:
1. closure property of [+]
2. closure property of [*]
3. additive identity property
4. additive inverse property
5. associative property of [+]
6. associative property of [*]
7. commutative property of [+]
8. left and right distributive property of [*] over [+]
Do you need a similar assignment done for you from scratch? We have qualified writers to help you. We assure you an A+ quality paper that is free from plagiarism. Order now for an Amazing Discount!
Use Discount Code "Newclient" for a 15% Discount!
NB: We do not resell papers. Upon ordering, we do an original paper exclusively for you.
