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Expert: - 7/24/2008


Question
Scott, thank you for the help with the other questions I asked. I have another abstract algebra question for you. Let Zn ={0,1,...,n-1} under addition and multiplication modulo n, for all n is a member of N where N stands for the set of all natural numbers, prove that Zn is a ring. Thank you, Richard.

Answer
The rules to be a ring are as follows:
Here we show these rules to be true.  If a=b, then a mod n = b mod n.
Using that, here we go.

1. Additive associativity: (a+b) mod n - (b+a) mod n sin we know that a+b=b+a.

2. Additive commutativity: ((a+b)+c) mod n = (a+(b+c)) mod n since we knoew that ((a+b)+c) = (a+(b+c)).

3. Additive identity: 0 is in the set and a+0=0+a.

4. Additive inverse: There exists an inverse of a and that is n-a for (a+n-a) mod n = n mod n = 0.

5. Left and right distributivity: (a*(b+c)) mod n = (ab + ac) mod n since a(b+c) = (ab + bc).

6. Multiplicative associativity: ((a*b)*c) mod n = (a*(b*c)) mod n since (a*b)*c = a*(b*c).

Therefore, the set is a ring.




Additional Information
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This was taken out of http://mathworld.wolfram.com/Ring.html:

Conditions 1-5 are always required. Though non-associative rings exist, virtually all texts also require condition 6 (Itô 1986, pp. 1369-1372; p. 418; Zwillinger 1995, pp. 141-143; Harris and Stocker 1998; Knuth 2000; Korn and Korn 2000; Bronshtein and Semendyayev 2004).

Rings may also satisfy various optional conditions:

a. Multiplicative commutativity: For all a, b, and c in the set S,
(a*b)*c = a*(b*c)  (a ring satisfying this property is termed a commutative ring),

b. Multiplicative identity: There exists an element 1 in the set S such that a*1=a (a ring satisfying this property is termed a unit ring, or sometimes a "ring with identity"),

c. Multiplicative inverse: For each a in S that is not 0, there exists an element a^-1 in S such that a*a^-1=a^-1*a=1, where 1 is the identity element.

Note that a and b are satisfied, but c is not.

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Scott A Wilson

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I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can even tell you it takes me over 2,000 steps to go a mile, but is that relevant?

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Experience in the area; I have tutored people in the above areas of mathematics for almost two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree almost 25 years ago. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

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My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

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Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

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I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

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My clients have been students at OSU, people nearby, friends with math questions, and several people every day on the PC, and you're probably make one more.

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