Description
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 3. 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 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 [+]
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.
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Explanation & Answer
The solutions are ready (docx and pdf files are identical). Please ask if something is unclear.Put particular attention to problems 1 and 2. All other are more or less beating the air (although they HAVE correct strict proofs).
Prove that ℤ𝑚 is a ring for any 𝑚. The operations are [𝑎] + [𝑏] = [𝑎 + 𝑏], [𝑎] ∙ [𝑏] = [𝑎 ∙ 𝑏].
[𝑐] means [𝑐]𝑚 , the equivalence class of 𝑐 modulo 𝑚.
1. Closure property of addition, [𝑎] + [𝑏] ∈ ℤ𝑚 .
Formally, [𝑎] + [𝑏] = [𝑎 + 𝑏] ∈ ℤ𝑚 as any equivalence class. But the problem is actually to prove that
adding classes by their representatives is a correctly defined operation, I.e. if we choose another
representatives of classes [𝑎], [𝑏], then the resulting class will be the same as [𝑎 + 𝑏].
Proof. Let [𝑎1 ] = [𝑎], [𝑏1 ] = [𝑏]. We need to prove that [𝑎 + 𝑏] = [𝑎1 + 𝑏1 ].
The equality [𝑐] = [𝑐1 ] means by definition that 𝑐 − 𝑐1 = 𝑚𝑞 for some integer 𝑞. Therefore
𝑎 − 𝑎1 = 𝑚𝑝,
𝑏 − 𝑏1 = 𝑚𝑞.
Thus (𝑎 + 𝑏) − (𝑎1 + 𝑏1 ) = (𝑎 − 𝑎1 ) + (𝑏 − 𝑏1 ) = 𝑚𝑝 + 𝑚𝑞 = 𝑚(𝑝 + 𝑞), which in turn means that
[𝑎 + 𝑏] = [𝑎1 + 𝑏1 ]. ∎
2. Closure property of multiplication, [𝑎] ∙ [𝑏] ∈ ℤ𝑚 .
Again, formally [𝑎 ∙ 𝑏] ∈ ℤ𝑚 but the correctne...
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