Rings Definition and 437 Threads

In Paul E. Bland's book: Rings and Their Modules, the author defines the external direct sum of a family of R-modules as follows: Two pages later, Bland defines the internal direct sum of a family of submodules of an R-module as follows: I note that in the definition of the external direct...

Homework Statement Determine all semisimple rings with a unique maximal ideal. The Attempt at a Solution If I call ##I## to the unique maximal ideal of ##R##, then ##I## can be seen as a simple ##R##-submodule, by hypothesis, there exists ##I' \subset R##, ##R-##submodule such that ##R=I...

For those looking for an undergraduate introduction to module theory there is not a great deal of choice regarding textbooks, but two possible texts are as follows: "A First Course in Module Theory" by M.E. Keating of Imperial College, London [Publisher: Imperial College Press, 1998] and...

Let R be a local ring with maximal ideal J. Let M be a finitely generated R-module, and let V=M/JM. Then if \{x_1+JM,...,x_n+JM\} is a basis for V over R/J, then \{x_1, ... , x_n\} is a minimal set of generators for M. Proof Let N=\sum_{i=1}^n Rx_i. Since x_i + JM generate V=M/JM, we have...

I'm trying to design mechanism similar to this video https://www.youtube.com/watch?v=t1JNWnTpmkA where i have concentric rings that spin within them selves around x, y and z axis. I intend to use track roller cam follower bearings to support each ring, with a bearing width of 10mm and a...

I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). I need help with Exercise 1.1.4 (iii) (Chapter 1: Basics, page 12) concerning matrix rings ... ... Exercise 1.1.4 (iii) (page 12) reads as follows: I can show that M_n (\alpha) is a...

I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). I need help with Exercise 1.1.4 (ii) (Chapter 1: Basics, page 13) concerning matrix rings ... ... Exercise 1.1.4 (ii) (page 13) reads as follows: I need help with Exercise (ii)...

I am reading Anthony W. Knapp's book, Basic Algebra. On page 146 in the section of Part IV (which is mainly on groups and group actions) which digresses onto rings and fields, we find the following text on the nature of ideals in the polynomial rings \mathbb{Q} [X] , \mathbb{R} [X] , \mathbb{C}...

Hello guys , Can someone explain me why color of central circle in transmission case is bright whereas it is dark for the reflection case? Thanks a lot,

am given that ϕ is a function from F(R) tp RxR defined by ϕ(f)=(f(0),f(1)) i proved that ϕ is a homomorphism from F(R) onto RxR. i showed that 1) ϕ(f) +ϕ(g)=ϕ(f+g) [for all f,g in F(R)] 2)ϕ(f)*ϕ(g)= ϕ(f*g) how do i show that ϕ is onto and define the kernal??(Wasntme)

I attempted to solve this problem by breaking up the semi-circle into infinitesimally small rings of mass dm and width dr a distance r away from the center [0<r<R]. I then wrote dm in terms of the area: dm=pi*r*dr. Then, I plugged into the formula di=integral(dm*r^2) and integrated from 0 to R...

I am reading Paul E. Bland's book, Rings and Their Modules. In Section 2.1: Direct Products and Direct Sums, Bland defines the direct product of a family of modules. He then, in Proposition 2.1.1 shows that there is a unique module homomorphism (or R-Linear mapping) from any particular R-module...

I am reading Paul E. Bland's book, Rings and Their Modules. In Section 2.1: Direct Products and Direct Sums, Bland defines the direct product of a family of modules. He then, in Proposition 2.1.1 shows that there is a unique module homomorphism (or R-Linear mapping) from any particular R-module...

So this is a review problem in our book I came across and i really want to understand it but I am just not having any luck, I did some research and found a guide on solving it but that's not really helping either. We didn't talk about unity elements in class and there aren't any examples in our...

Homework Statement These are problems from a Newton rings experiment where a lens was placed on a flat surface and the interference patterns created Newton rings. I measured the diameter of the first five rings and then plotted a graph of d^2 against N (number of the individual ring). These...

What is the lower mass limit for a body to have rings? I'm thinking that Chariklo, at 155 miles in diameter is pretty small. Gravity would be so weak you'd have to hold on just to not 'jump' off with every step. Cheers, 3point14rat

http://webcache.googleusercontent.com/search?q=cache%3Ahttp%3A%2F%2Fwww.wral.com%2Fdiscovery-an-asteroid-with-rings%2F13510966%2F "The European Southern Observatory announced this week the discovery of rings around an asteroid 1.4 billion miles from Earth. Chariklo is the largest of the class...

My Son is in 5th Grade and currently his class is studying the solar system. he wanted to know why only Gas giants had rings and the smaller planets did not. His Teacher apparently side stepped the question so he asked me. I assumed it is due to the Gravity but I didn't have enough information...

Apologies for a long preamble. There is a simple question at the end of this post. Wikipedia and the PF archive were not forthcoming on this subject. Saturn's rings seem sharply bounded in the vertical dimension. If I got it right, the initial orbits of debris were centered on the orbits of...

Hi everyone, :) Here's another question that I am struggling to complete. If you have any hints or suggestions for this one, I would be so grateful. :) Question: Let $S\subseteq R$ be rings and assume that $R_S$ is a finitely generated $S$-module. If $S$ is Artinian prove that $R$ is also...

Hi everyone, :) Here's a question that I am struggling find the answer. Any nudge in the correct direction would be greatly appreciated. Question: Prove that a prime Artinian ring is simple.

Homework Statement Hello. I have this examples about Saturn: A) Calculate the radius of the inner edge of the Cassini division, if you know that this sharp transition in the density of the rings is caused by a 2:1 orbital resonance with the moon ring particles with a moon Mimas, which around...

To guard against buckling modes I know that circumferential stiffeners can strengthen a cylindrical vessel against external pressure / internal vacuum. Question: Stiffer rings do not contribute anything to protect vessel against internal pressure, correct? I think not, based on Stress eqs...

Hi everyone, :) I find it difficult to get the exact meaning of the following question. What does "in a natural way" means? Is it that we have to show that there exist a isomorphism between \(S\) and a sub-module of \(R\)? Any ideas are greatly appreciated. :) Question: Given a ring...

I am reading R Y Sharp: Steps in Commutative Algebra. In Chapter 3 (Prime Ideals and Maximal Ideals) on page 44 we find Exercise 3.24 which reads as follows: ----------------------------------------------------------------------------- Show that the residue class ring S of the ring of...

I am reading R.Y. Sharp: Steps in Commutative Algebra. Lemma 1.13 on page 7 (see attachment) reads as follows: -------------------------------------------------------------------------------------- 1.13 LEMMA. let R be a commutative ring, and let X be an indeterminate; let T be a commutative...

I am reading Dummit and Foote Chapter 15, Section 15.1: Noetherian Rings and Affine Algebraic Sets. Exercise 10 reads as follows: -------------------------------------------------------------------------------------------------------------------- Prove that the subring: k[x, x^2y, x^3y^2...

I am reading Dummit and Foote Chapter 15, Section 15.1: Noetherian Rings and Affine Algebraic Sets. Exercise 9 reads as follows: ------------------------------------------------------------------------------------------------ For k a field show that any subring of a polynomial ring k[x]...

I am reading R. Y. Sharp: Steps in Commutative Algebra, Chapter 5 - Commutative Noetherian Rings Exercise 8.5 on page 147 reads as follows: -------------------------------------------------------------------------------------------------- 8,5 Exercise. Show that the subring \mathbb{Z} [...

Homework Statement See attached image Homework Equations The Attempt at a Solution For the first half of the question, ordered pairs would be (1, [1]), since 1 and [1] are the multiplicative identities in these rings. but no matter how many times we add (1, [1]) to itself...

there are 2 rings of charge, radius R on the x-axis separated by a distance R, find the potential and E field. so don't you have to calculate the field to the left of the 2 rings, in between the 2 rings and to the right? I get these answers for the E field which by symmetry points only on the...

In Dummit and Foote Chapter 13: Field Theory, the authors give several examples of field extensions on page 515 - see attached. In example (3) we read (see attached) " (3) Take F = \mathbb{Q} and p(x) = x^2 - 2 , irreducible over \mathbb{Q} by Eisenstein's Criterion, for example" Now...

In Dummit and Foote Chapter 15 Exercise 2(a) on page 668 reads as follows: Show that the following ring is not Noetherian by exhibiting an explicit infinite increasing chain of ideals: - the ring of continuous real valued functions on [0, 1]I would appreciate help on this exercise. Peter...

Dummit and Foote Exercise 1 on page 668 states the following: "Prove the converse to Hilbert's Basis Theorem: if the polynomial ring R[x] is Noetherian then R is Noetherian" Can someone please help me get started on this exercise. Peter [Note: This has also been posted on MHF]

I am reading R.Y Sharp's book: "Steps in Commutative Algebra". On page 6 in 1.11 Lemma, we have the following: [see attachment] "Let S be a subring of the ring R, and let \Gamma be a subset of R. Then S[ \Gamma ] is defined as the intersection of all subrings of R which contain S and...

In Exercise 2 of Exercises 1-5 of John Dauns' book "Modules and Rings" we are given R = \begin{pmatrix} \mathbb{Z}_4 & \mathbb{Z}_4 \\ \mathbb{Z}_4 & \mathbb{Z}_4 \end{pmatrix} and M = \overline{2} \mathbb{Z}_4 \times \overline{2} \mathbb{Z}_4 where the matrix ring R acts on a right...

In the book "Modules and Rings" by John Dauns he adopts the following notation in Exercises 1-5 (see attached) "For additive groups A, B, C, D and a \in A, b \in B, c \in C, d \in D we write [a,b; c,d] = \begin {vmatrix} a & b \\ c & d \end {vmatrix} ; [A, B, C, D] = \begin {vmatrix} A...

Dummit and Foote in Chapter 9 - Polynomial Rings define Noetherian rings as follows: Definition. A commutative ring R with 1 is called Noetherian if every ideal of R is finitely generated. Question: Does this mean that R itself must be finitely generated since R is an ideal of R? This...

May I ask what are some of the reasons we see so much aromatic, hetrocyclic chemistry in biology? I realize that several amino acids are aromatic and so to nucleic acids, other biomolecules. What exactly do the aromatic rings bring to biology in order for us to see it so profusely utilized?

Okay, So I just got done reading and re-reading Chapter one of Introductory Real Analysis by Andrei Kolmogorov and S.V Fomin, I have to say it is a lot easier to read than the symbolic logic book I have. But to the point, There is this section which talks about Boolean Rings and it defines them...

I just recently found a brilliant lectures on rings which cleared so many thing up, and opened a whole deal, or prime Ideal(lol) of insights for me, But for now I will work and develop those insights and for the mean time I wish to show you these video, and hope you find it interesting. Link \\...

When we write F[x_1, x_2, ... ... , x_n] where F is, say, a field, do we necessarily mean the set of all possible polynomials in x_1, x_2, ... ... x_n with coefficients in F? [In this case, essentially all that is required to determine whether a polynomial belongs to F[x_1, x_2, ... ... ...

Let’s also say that 65% of all diamond rings sold in WV are yellow gold. In a random sample of 20 folks, what is the probability that 12 have purchased yellow gold diamond rings?thank you and I promise this is the last question today.:D

I am reading Dummit and Foote Section 9.5 Polynomial Rings Over Fields II. I am trying to understand the proof of Proposition 16 which reads as follows: --------------------------------------------------------------------------------- Proposition 16. Let F be a field. Let g(x) be a...

I am reading Dummit and Foote Section 9.5 Polynomial Rings Over Fields II and need some help and guidance with the proof of Proposition 15. Proposition 15 reads as follow: Proposition 15. The maximal ideals in F[x] are the ideals (f(x)) generated by irreducible polynomials. In particular...

Homework Statement Is there a finite non-trivial ring such that for some a, b in R, ac = bc for all c in R? Does there exist finite non-trivial rings all of whose elements are zero-divisors or zero? 2. The attempt at a solution Let a, b ≠ 0 in R such that ac=bc for all c in R...

Could the formation of rings occur on a planet which is tidal locked?

On page 222 of Nicholson: Introduction to Abstract Algebra in his section of Factor Rings of Polynomials Over a Field we find Theorem 1 stated as follows: (see attached) Theorem 1. Let F be a field and let A \ne 0 be an ideal of F[x]. Then a uniquely determined monic polynomial h exists...

I am reading Dummit and Foote Section 9.4 Irreducibility Criteria. In particular I am struggling to follow the proof of Eisenstein's Criteria (pages 309-310 - see attached). Eisenstein's Criterion is stated in Dummit and Foote as follows: (see attachment) Proposition 13 (Eisenstein's...

I am reading Dummit and Foote Section 9.3 Polynomial Rings That are Unique Factorization Domains (see attachment Section 9.3 pages 303 -304) I am working through (beginning, anyway) the proof of Theorem 7 which states the following: "R is a Unique Factorization Domain if and only if R[x] is a...