ASSUMPTIONS A set, R, with two operations,+ and *, is a ring if the following properties...

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ASSUMPTIONS A set, R, with two operations,+ and *, is a ring if the following properties are shown to be true: 1. Closure property of addition: for all s and tin R, s+ tis also in R 2. Closure property of multiplication: for all sand tin R, s* t is also in R 3. Additive identity property: there exists an element O in R such that s+ 0 =s for all sin R 4. Additive inverse property: for every sin R, there exists tin R, such that s+ t=0 5. Associative property of addition: for every q, s, and tin R, q+ (s+t) = (q+s) +t 6. Associative property of multiplication: for every q, s, and tin R, q*(s*t)=(a*s)*t 7. Commutative property of addition: for all sand tin R, s+t=t+s 8. Left distributive property of multiplication over addition: for every q, s, and tin R, q*(s+t)=q*s+q*t 9. Right distributive property of multiplication over addition: for every q, s, and tin R, (s+t)*q=s*q+t*q Given the set of integers mod m denoted Zm, the elements of Zmare denoted [x]m, where x is an integer from 0 to m - 1. Each element [x]mis an equivalence class of integers that has the same integer remainder as xwhen divided by m. Consider, for example, Z7 = {[0]7, [1], [2]7, [3]7, [4]7, [5]7, [6]7}. The element [5]7 represents the infinite set of integers of the form 5 plus an integer multiple of 7. That is, [5]7={...-9, -2, 5, 12, 19, 26,...}, or, more formally, [5]7={y: y=5+7q for some integer q}. Modular addition, +, is well defined on the set Zm in terms of integer addition as follows: [a]m+ [b]m=[a+b]m Modular multiplication,*, is well-defined on the set Zmin terms of integer multiplication as follows: [a]m* [b]m= [a* b]m The set of integers Zforms a ring with the usual operations of integer addition and multiplication. Given this fact, you are asked to prove that Zm for an assigned value of malso has properties of a ring in part A of this task. Each step of each proof must be justified using an appropriate property from the ring Zor the given definitions of modular arithmetic operations. ASSUMPTIONS A set, R, with two operations,+ and *, is a ring if the following properties are shown to be true: 1. Closure property of addition: for all s and tin R, s+ tis also in R 2. Closure property of multiplication: for all sand tin R, s* t is also in R 3. Additive identity property: there exists an element O in R such that s+ 0 =s for all sin R 4. Additive inverse property: for every sin R, there exists tin R, such that s+ t=0 5. Associative property of addition: for every q, s, and tin R, q+ (s+t) = (q+s) +t 6. Associative property of multiplication: for every q, s, and tin R, q*(s*t)=(a*s)*t 7. Commutative property of addition: for all sand tin R, s+t=t+s 8. Left distributive property of multiplication over addition: for every q, s, and tin R, q*(s+t)=q*s+q*t 9. Right distributive property of multiplication over addition: for every q, s, and tin R, (s+t)*q=s*q+t*q Given the set of integers mod m denoted Zm, the elements of Zmare denoted [x]m, where x is an integer from 0 to m - 1. Each element [x]mis an equivalence class of integers that has the same integer remainder as xwhen divided by m. Consider, for example, Z7 = {[0]7, [1], [2]7, [3]7, [4]7, [5]7, [6]7}. The element [5]7 represents the infinite set of integers of the form 5 plus an integer multiple of 7. That is, [5]7={...-9, -2, 5, 12, 19, 26,...}, or, more formally, [5]7={y: y=5+7q for some integer q}. Modular addition, +, is well defined on the set Zm in terms of integer addition as follows: [a]m+ [b]m=[a+b]m Modular multiplication,*, is well-defined on the set Zmin terms of integer multiplication as follows: [a]m* [b]m= [a* b]m The set of integers Zforms a ring with the usual operations of integer addition and multiplication. Given this fact, you are asked to prove that Zm for an assigned value of malso has properties of a ring in part A of this task. Each step of each proof must be justified using an appropriate property from the ring Zor the given definitions of modular arithmetic operations.