Abstract algebra Definition and 458 Threads

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states...

Below in the quoted passage is expressing the definition of group action in terms of the definition of a group and vice versa. Can someone check if it there any mistakes please. I had helped with one of the LLMs. I always want to know for the two definitions, if one can be written in terms of...

##\textbf{Definition 1:}## Let ##R## be an UFD. A nonzero polynomial in ##R[x]## is said to be ##\textbf{primitive}## if the only constants that divides it are the units in ##R##. ##\textbf{Definition 2:}## Let ##R## be an UFD. Hence highest common factors of finite subsets of ##R## exists...

Let ##R## be a UFD. If ##a\mid bc## and ##1_R## is a gcd of ##a## and ##b,## prove that ##a\mid c.## Solution: Let ##\text{gcd}(a,b)=1_R##, then ##(a,b)\sim 1_R.## Then there exists ##x,y\in R## so that ##ax+by=1_R##. So ##c(ax+by)=c## for some ##c\in R.## Since ##a\mid bc,## implies...

Questions For the two passages in the Background below highlighted in bold, I am having trouble figuring how the maps are defined. I understand that we have commutative ring ##R## admitting a total quotient ring ##F##, a, subset ##M\subset R## which is also a regular multiplicative system. A...

The following is from the paper "A Group Epimorphism is Surjective" by C.E. Linderholm Background Note: where it says the function from ##A/H## to ##A/H##, should it not be ##H/A## to ##H/A?## Questions There are some points about the above proof I am not clear on. I understand that...

The following is taken from Abstract Algebra: A First Course by Stephen Lovett Background Exercise Let ##D=\{2^a 3^b\mid a,b,\mathbb{N}\}## as a subset of ##\mathbb{Z}##. Prove that ##D^{-1}\mathbb{Z}## is isomorphic to ##\mathbb{Z}[\frac{1}{6}]## even though ##D\neq\{1,6,6^2,\ldots\}##...

Background I have questions about localizing the following quotient polynomial ring ##R=\frac{\mathbb{Z}[x]}{(x^2+1)(x^5+11x^2+3)}## at the prime ideal ##M=(x^5+11x^2+3)## Question: For the question above, my attempted solution is as follows: Let ##I=((x^2+1)(x^5+11x^2+3))## and let ##S##...

I asked online questions about Proposition 2.1.1: The answer I got is the following: I have some questions about the answer I got. When the person answering says: ##1.## Is the map ##\mathfrak{q}\mapsto \mathfrak{q} A _\mathfrak{p}## from ##A\setminus \mathfrak{p}\to...

The following question is taken from ##\textit{Arrows, Structures and Functors the categorical imperative}## by Arbib and Manes. ##\color{Red}{Questions:}## Is the follow a correct concrete example for the coproduct of ##\textbf{Abm},## (category of abelian monoid) by modifying notations as...

The following is taken from a chapter from Algebraic geometry and commutative algebra by S Bosch Background and ##\mathrm{Spec}R=\{\mathfrak{p}\subset R:\mathfrak{p}\text{ prime ideal in }R\}##. For reference, I included a screenshot of the typed out quoted passage above below: Questions...

The Definitions below are taken from the following books: Fundamentals of abstract algebra by: Malik, Morderson, Sen Rings, Modules and Algebras by: Adamson First Course in Module Theory by: Keating Basic Abstract Algebra by: Bhattacharya, Jain Nagpaul How to Prove it by Dan Velleman...

This post is a further clarification of this post here; Notation questions about kernel, cokernel, image and coimage In that post, I asked about notations related difficulties about cokernel, coimage, image, kernel. I did some more research, and I am more clear on how to articulate what I am...

The following are taken from Lectures on Algebra by Shreeram Shankar Abhyankar Commutative Algebra Volume 2 By Oscar Zariski, Pierre Samuel Abstract Algebra An Introduction by Thomas Hungerford Concepts in Abstract Algebra by Charles Lanski Background Questions The passages quoted...

The following are taken from Locally Compact Groups by Markus Stroppel Commutative Algebra Volume 1 By Oscar Zariski, Pierre Samuel Introduction to the Algebraic Geometry and Algebraic Groups Volume 39 by Michel Demazure Basic Algebra by Anthony Knapp General Topology by J. Dixmier...

The two screenshots below are taken from the book Fundamentals of Modern Algebra A global Perspective by: Robert G Underwood Screenshot 1 Screenshot 2 The two screenshots above concerns localizing a non integral domain at a prime; in the second screen shot, the example given is ##(Z_6)_3##...

The screenshots below are taken from the two books Algebra: Chapter 0 by: Paolo Aluffi and Abstract Algebra A comprehensive Introduction by Lawrence and Zorzitto. The first two screenshots are from Aluffi's text, while the last four are from Lawrence and Zorzitto. Screenshot 1 Screenshot 2...

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that...

The screenshots below are taken from the 2nd editon of the book How to Prove it A structured approach By: Daniel Velleman and and 3rd edition of the book's solution manual. (Page 5) The question on page 4 exercise 5b corresponds to the solution in page 5 exercise 6b Page 1 Page 2 Page 3...

Background Notation: ##\Bbb{Q}_{\Bbb{Z}}[x]=\Bbb{Z}+x\Bbb{Q}[x]## Assumed exercises: (1)(a) Prove that the only units in ##\Bbb{Q}_{\Bbb{Z}}[x]## are ##1## and ##-1##. (b) If ##f(x)\in \Bbb{Q}_{\Bbb{Z}}[x]##, show that the only associates are ##f(x)## and ##-f(x)##. (2)(a) If ##p## is...

This is a pattern I noticed when playing around with Mathematica. Is there any way to rigorously prove this? I was not able to find any literature concerning the number of factors in a finite field, especially because this is called a "pentanomial" in said literatures. These don't have much...

The following are from Froberg's "Introduction to Grobner bases" , and Hungerford's undergraduate "Abstract Algebra" text, and also a continuation of this...

The Math challenge threads have returned! Rules: 1. You may use google to look for anything except the actual problems themselves (or very close relatives). 2. Do not cite theorems that trivialize the problem you're solving. 3. Do not solve problems that are way below your level. Some problems...

Hello everyone, I am an International Baccalaureate (IB) student working on my extended essay, which is a mandated 4,000-word research paper. My chosen topic is Quaternions, a mathematical concept I find highly intriguing. The primary aim of my paper is to model the rotation of an asteroid...

picture since the text is a little hard to read i have no problem showing this is a vector space, but what is meant by complex dimention? Is it just the number on independant complex numbers, so n?

Let ## \varphi \subseteq A \times B; \psi \subseteq B \times C ##. Then ## \varphi \circ \psi = \left \{ (a, c)| \exists b: (a,b) \in \varphi, (b,c) \in \psi \right \} \subseteq A \times C##. Task: Let ##\varphi## and ##\psi## are subalgebras of algebras ##A \times B## and ##B \times C##...

...Out of interest am trying to go through the attached notes, My interest is on the highlighted, i know that in ##\mathbb{z}/\mathbb{6z}## under multiplication we shall have: ##1*1=1## ##5*5=1## am assuming that how they have the ##(\mathbb{z}/\mathbb{6z})^{*}={1,5}## is that correct...

I'm used to seeing commutative diagrams where the vertices are mathematical objects and the edges (arrows) are mappings between them. Can the diagram ( from the interesting article https://people.reed.edu/~jerry/332/25jordan.pdf ) in the attached photo be interpreted that way? In the...

Hello, I have a question that I would like to ask here. Let ##L = \left\{ x \in \mathbb{Z}^m : Ax = 0 \text{ mod } p \right\}##, where ##A \in \mathbb{Z}_p^{n \times m}##, ##rank(A) = n##, ## m \geq n## and ##Ax = 0## has ##p^{m-n}## solutions, why is then ##|L/p\mathbb{Z}^m| = p^{m-n}##? I...

Please, I have a question about automorphism: Let ##\mathbb{K}## be a field, if ##\operatorname{char}(\mathbb{K})=p ##, then the order of automorphism ##\phi## is ##p##, i.e. ##\phi^p=\operatorname{id}##, where ##i d## is identity map. Is that right? please, if yes, how we can prove it, and...

I did not use the hint for this problem. Here is my attempt at a proof: Proof: Note first that ##σ(σ(x)) = x## for all ##x \in G##. Then ##σ^{-1}(σ(σ(x))) = σ(x) = σ^{-1}(x) = σ(x^{-1})##. Now consider ##σ(gh)## for ##g, h \in G##. We have that ##σ(gh) = σ((gh)^{-1}) = σ(h^{-1}g^{-1})##...

Hi all, It is nice to be a member in this site! Hope it will be beneficial and add to my knowledge and understanding.

I would wish to receive verification for my proof that ##sup\{a \in \mathbb{Q}: a^2 \leq 3\} = \sqrt{3}##. • It is easy to verify that ##A = \{a \in \mathbb{Q}: a^2 \leq 3\} \neq \varnothing##. For instance, ##1 \in \mathbb{Q}, 1^2 \leq 3## whence ##1 \in A##. • We claim that ##\sqrt{3}## is an...

Hello, I am looking for one or more books in combination for self-study of abstract algebra. Desirable would be a good structure of the book with good examples of sentences and definitions. Of course, exercise problems should not be missing. I am now almost tending to buy the Algebra 0 book by...

I would like to show that a LLL-reduced basis satisfies the following property (Reference): My Idea: I also have a first approach for the part ##dist(H,b_i) \leq || b_i ||## of the inequality, which I want to present here based on a picture, which is used to explain my thought: So based...

Hello, I've been thinking a bit about the definition of the ##i##-th successive minima of a lattice (denoted with ##\lambda_i(\Lambda)##), and I would argue that the ##i##-th successive minimum is at most as large as the largest lattice basis vector ##b_i##. More formally...

School starts soon, and I know students are looking to get their textbooks at bargain prices 🤑 Inspired by this thread I thought that I could share some of my findings of 100% legally free textbooks and lecture notes in mathematics and mathematical physics (mostly focused on geometry) (some of...

let p∈Z a prime how can I show that p is a prime element of Z[√3] if and only if the polynomial x^2−3 is irreducible in Fp[x]? ideas or everything is well accepted :)

please offer me examples of: a) 3 vector spaces over the same field; and b) the same vector space over 3 fields.

I read this article History of James Clerk Maxwell and it talks about Maxwell and Dirac also at some point. It is said that Maxwell thought geometrically, and also Dirac said he thought of de Sitter Space geometrically. They say their approach to mathematics is geometric. I see this mentioned...

I'm following this video on how to establish an equivalence relation to define the tensor product space of Hilbert spaces: ##\mathcal{H1} \otimes\mathcal{H2}={T}\big/{\sim}## The definition for the equivalence relation is given in the lecture vidoe as ##(\sum_{j=1}^{J}c_j\psi_j...

I believe that I am correct, the following statement here must be FALSE, right? It has to be false because A union B is like the two entire circles of the Venn diagram and that cannot be a subset of the intersection area, right? Now if this statement was flipped, then it would be true?

Dear Everyone, What are the strategies from proving a either-or statements? Is there a way for me to write an either-or statement into a standard if-then statements? For example, this exercise is from Dummit and Foote Abstract Algebra 2nd, "Let $x$ be a nilpotent element of the commutative...

My university offers two different two-semester sequences for learning abstract algebra, and I can't decide which one would be better for me, a physics major. Here are the two sequences and their course descriptions, copied and pasted from the university website: Algebra 1: Theory of groups...

Homework Statement Find all cosets of the subgroup H in the group G given below. What is the index (G : H)? H = <(3,2,1)>, G = S3 Homework EquationsThe Attempt at a Solution I will leave out the initial (1,2,3) part of the permutation. We have S3 =...

One of the last classes I'm taking before finishing my degrees as an undergraduate is abstract algebra. My professor uses the textbook 'Contemporary Abstract Algebra' by Joseph Gallian. The book isn't written terribly nor is the teacher a poor one, but I just find this subject so...

<Moderator's note: Moved from a technical forum and thus no template. Also re-edited: Please use ## instead of $$.> If ##R_{1}## and ##R_{2}## are relations on a set S with ##R_{1};R_{2}=I=R_{2};R_{1}##. Then ##R_{1}## and ##R_{2}## are bijective maps ##R_{1};R_{2}## is a composition of two...

This time my struggle is with ring ideals. Book still won't provide examples, so I'm again trying to come up with some of my own. I figured {0,2} might fit the definition as an ideal of ##\mathbb{Z/4Z}## since it is an additive subgroup and ##\forall x \in I, \forall r \in R: x\cdot r, r\cdot x...

So I'm just beginning to study abstract algebra and I'm not sure I grasp the definition of a quotient group, I believe it probably has to do with the book providing little to no examples. In trying to come up with my own examples, I imagined the following: Consider the Klein four group, if we...

I am reading The Basics of Abstract Algebra by Paul E. Bland ... I am focused on Section 3.2 Subrings, Ideals and Factor Rings ... ... I need help with another aspect of the proof of Theorem 3.2.19 ... ... Theorem 3.2.19 and its proof reads as follows...