# rings

1. ### Symmetries when expanding Thue Morse sequence in layers of rings

By generating Thue Morses sequence in rings and study the natural numbers N (including 0) represented by radial binary combinations some geometrical properties emerges, such as: * All odd integers will be arranged in a specific geometric order * Even integers will be arranged in a specific...
2. ### Rings Problem

Hello all, I have done part (a) of the question as attached and am not sure if they are correct. Would appreciate if you can help me to see. Next, I have no idea how I should do part (b). Greatly appreciate! Thanks in advance! Axiom 4 and 5 are found in the written image.
3. ### Construction of a polynomial rings

Can anyone explain why we need to construct a polynomial ring necessarily over a commutative ring with an identity? What would happen if we tried to make one over a noncommutative ring without identity?
4. ### One theorem for rings

Let R be finite, commutative ring with 1=/=0. Statements below are equivalent: a) a \in U(R), where U(R) is a set of invertible elements of R b) a is not a zero divisor c) a^m=1, where m is the number of elements that are not zero divisors in R My task is to discuss this theorem in terms...
5. ### Rings with four elements

Good morning. :) Recently I've been doing some research on a ring theory. I found out that there 11 rings that have four elements (is the number of them correct or are they less or more?)? Here comes my question: why are there only 11 of them? Can someone explain this in a language of abstract...
6. ### Quotient Rings

I am having trouble understanding ideals. If R is a commutative ring, and I an ideal of R, can someone please explain what an ideal of R/I looks like? Thank you.
7. ### A few cheap safe power sources to obsolete fusion and lead to dyson rings

Harness the difference in electricity between ocean top and many miles below, which MUST be hugely different because of all the solar power diffused into the TOP of the ocean and not yet had time to get below, but it will average over time, so ALTERNATING CURRENT will provide at least enough to...
8. ### Rings

Help with this test please. Thank you !!!
9. ### Dedekind Domains - Discrete Valuation rings

An integral domain A is a discrete valuation ring if and only if (a) A is Noetherian (b) A is integrally closed, and (c) A has exactly one nonzero prime ideal. Can someone prove this please?
10. ### Groups, rings and fields

I'm studying a course about linear algebra and it starts with groups fields and rings. Now I there are some example questions which ask me why the following things are true: K is a field. Why is K not a group in multiplication? R is a ring. Why is R\ {0} not a group in multiplication ...
11. ### Structure of rings

Dear friends, could you help me please for the 1st and 2nd questions please?
12. ### Structure of rings

Dear friends, could you help me to solve for the 1st and 2nd questions please?
13. ### Rings. Z4[x]

My kind regards to you I am thinking about rings. But what does Z4[x] and Z5[x] mean? If you calculate something, I don't know how you should use those. I think that Z4[x] contains 0,1,2,3 and Z5[x] contains 0,1,2,3,4. But I don't understand more about those. Can you give some examples?
14. ### Rings

Hi , i have this problem If n, m in Z , with "n" divisor of "m" . Prove that the natural projection of rings Z/mZ ------> Z/nZ defined by x+mZ ---> x+nZ is also surjective in the units : (Z/mZ)* ------>( Z/nZ)*
15. ### Rings and Subrings

Hey! Ran into a question i'm not quite getting. Let R be a ring and let S and T be subrings of R. Show that S\cap T is also a subring of R. This obviously makes sense but I'm just not sure where to start. Any help would be great!
16. ### 2 Rings of Polynomials Questions

I have two separate questions: (1) I want to know if x^8 +1 = x^3 + 1 is in \mathbb{Z}_5[x]? Is this one a simple counter example problem? If so, let x = 2 which is in Z_5. I am assuming that the problem means that both x^8+1 and x^3+1 are in \mathbb{Z}_5[x]. 2^8 + 1 = 2^3+1 257 = 9 257 is...
17. ### Difference between rings and fields.

What the difference between 'ring' to 'field'? By which way they are different from one another?
18. ### Nilpotent elements in rings

Can anyone please help with the following problem from Dummit and Foote: Abstract Algebra Ch 7 If a is an integer, show that the nilpotent element \overline{a} \in \mathbb{Z} /n \mathbb{Z} is nilpotent if and only if every prime divisor of n is also a divisor of a. In particular, determine...
19. ### projective dim over local rings

It is a routine fact about modules over commutative, unital, Noetherian local ring \left ( R \, , \mathfrak{m} \, , k \right ) that: For any finitely generated R-module M, we have pd M = \sup \left \{ i \in \mathbb{N}_0 : \, {\rm Ext}^i \left ( M \, , k \right ) \ne 0 \right \}, where pd M...
20. ### Polynomial Rings

Let F be a field, and let f(x) and g(x) belong to F[x]. If there is no polynomial of positive degree in F[x] that divides both f(x) and g(x) [in this case f(x) and g(x) are said to be relatively prime], prove that there exist polynomials h(x) and k(x) in F[x] with the property that f(x)h(x) +...