Random Ring theory Questions

In summary, Z/(p) is a field, Euclidean Domain, Principal Ideal domain, Unique factorization domain, and Domain. Z/(a) is not a field, but it is a Noetherian. R[x] is not a field, but it is a Euclidean domain and a Noetherian. R[x,y] is not a field, but it is a Noetherian. Different rings have different properties, and it is important to understand these distinctions for a general understanding of rings.
  • #1
nadineM
8
0
I know that Z/(p) that is the integers mod a prime ideal is a field
and I also know that:
Field -> Euclidean Domain -> Principal Ideal domain -> Unique factorization domain ->Domain
So I know that Z/(p) are all of these things.
I also know that Z/(a) That is the set of integers mod a non prime number is not a field. But is it any of the other things? That is it a Euclidean Domain, Principal Ideal domain, Unique factorization domain,Domain, or noetherian?
I have the same question about R[x] and R[x,y] By this notation I mean the sets of polinomials with real coefficients in one and two variable. I believe that R[x] has all of the above listed properties. But I was wondering about R[x,y] I beilve it is not a field but it is noetherian, but I don't know about the other properties...

Can anyone clear these things up? I am just trying to come to a more general understanding of what properties are the same and different in different rings. Thanks
 
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  • #2
Z/(a) is not a domain when a is composite, and this is very easy to prove, so do it. Since these rings are finite, they are Noetherian because there are only finitely many subsets of these rings, hence only finitely many ideals (so any increasing chain of ideals must certainly stabilize).

R[x] is not a field, what would the inverse of x be? It is a Euclidean domain though. The Euclidean algorithm for polynomial division is something you should have learned in high school, it's just long division. Both R[x] and R[x,y] are Noetherian according to Wikipedia.
 
  • #3


It is correct that Z/(p) is a field, as it satisfies all the properties listed (Euclidean Domain, Principal Ideal domain, Unique factorization domain, Domain). However, Z/(a) where a is a non-prime number is not a field and it also does not satisfy any of the other properties listed. It is not a Euclidean Domain, Principal Ideal domain, Unique factorization domain, Domain, or noetherian. This is because Z/(a) is not an integral domain, meaning it has zero divisors, and it does not have the property of unique factorization of elements into irreducible elements.

As for R[x] and R[x,y], R[x] is indeed a field and satisfies all the properties listed. However, R[x,y] is not a field and it is not a Euclidean Domain or a Principal Ideal domain. It is a Unique factorization domain and a Domain, but it is not noetherian. This is because R[x,y] is not a principal ideal domain, meaning not all ideals can be generated by a single element.

In general, it is important to note that different rings can have different properties and it is not always possible to generalize. It is important to understand the specific properties and characteristics of each individual ring.
 

1. What is Random Ring theory?

Random Ring theory is a branch of mathematics that studies the properties and behaviors of rings, which are algebraic structures consisting of a set of elements and two binary operations, addition and multiplication. The "random" aspect refers to the use of probability and randomness in studying these structures.

2. How is Random Ring theory used in science?

Random Ring theory has applications in various fields such as physics, computer science, and statistics. In physics, it can be used to model random processes and systems. In computer science, it has applications in coding theory and cryptography. In statistics, it is used in the study of random variables and stochastic processes.

3. What are some key concepts in Random Ring theory?

Some key concepts in Random Ring theory include ring homomorphisms, ideals, and fields. A ring homomorphism is a function between two rings that preserves the ring structure. Ideals are subsets of a ring that behave like "multiples" of an element in the ring. A field is a ring in which every non-zero element has a multiplicative inverse.

4. How does Random Ring theory differ from traditional Ring theory?

Random Ring theory differs from traditional Ring theory in that it incorporates randomness and probability into the study of rings. Traditional Ring theory focuses on the algebraic properties and structures of rings, while Random Ring theory also considers the probabilistic behavior of these structures.

5. What are some open problems in Random Ring theory?

Some open problems in Random Ring theory include the study of random rings with infinite elements, the classification of random rings, and the relationship between random rings and other areas of mathematics such as group theory and topology. Other open problems include the development of efficient algorithms for computing with random rings and the application of Random Ring theory to real-world problems.

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