Modules Definition and 214 Threads

Definition. Let ##F## be a field and ##G## be a group. An ##FG##-module is a finite-dimensional vector space ##V## on which ##G## acts (from the right: ##V \times G \to V, (v,g)\mapsto v\cdot g##) such that the next conditions hold: 1) ##(v \cdot g)\cdot h = v \cdot (g \cdot h)## 2) ##v \cdot e...

I have this remote server where I loaded the ISP-provided OS, namely Ubuntu 22.04. The lsb_release -d shows "Ubuntu 22.04.4 LTS" and uname -r shows "5.2.0". My problem arose when there seemed to be missing modules for kernel 5.2.0 in /lib/modules/5.2.0 for my needs. There is also no...

Let ##M## be a flat module over a commutative ring ##A##. Suppose ##X_1## and ##X_2## are submodules of an ##A##-module ##X##. Prove that ##(X_1 \cap X_2) \otimes_A M = (X_1 \otimes_A M) \cap (X_2 \otimes_A M)## as submodules of ##X\otimes_A M##.

My wife asked me a good question, and even with moderate Google searching I couldn't answer her question about how the patches are installed in arms. How do these BGL monitoring patches work? Thanks...

I'm figuratively beating myself up for not knowing about these modules when I went to write my modules. On one hand, I feel like it would be a giant hassle to rewrite them just to implement some module. One of these modules is over a thousand lines long, which might be inconsequential to...

I am working on Windows 10 and using VSCode. In my project, the folder/file tree looks like this; \Equations __init__.py equation_producer.py \Objects __init__.py \GRTensors __init__.py metrictensor.py riccitensor.py riemanntensor.py ...Now I want to...

Hello, I have been trying to figure out where things are stored in Windows. For example, I have two versions of Python, one is stored in the PYthon37 folder and one in the Anaconda3 folder. See below: I have been using pip at the command prompt to install packages/modules and when typing at...

I'm working with Python modules at the mo', and I am having trouble trying to decide how best to include supplementary material that is accessed by my module, but not necessarily a part of it. The program works alright when accessing it from the IDE, but it fails to recognize the files when I'm...

Hello, I understand that modules are essentially Python file save as .py. These files contain both functions and/or classes. To use them in our programs, we must use the keyword import. However, this works only if the module is available, i.e. already installed in the standard Python library...

In 2D modules, the 3rd direction isn't shown in model settings. What assumptions are made regarding electrostatics 2D modules? For example, how is a 2D Poisson's equation with point sources solved? Is it based on a 1/r potential or a log potential?

In AC-DC power supply modules, having non-isolated power converter topology may result in input "Line" (hot wire) been assigned to "Vss" (lower voltage rail) at output of power supply module, and N (neutral) assigned to Vdd (upper power rail). Inside single system, this usually does not results...

I have a question regarding the minimum operating speed of new SFP+ fiber modules. In my line of work, I'm most commonly seeing 10GBASE SFP+ modules installed for network data. Can these modules operate at low speed? Specifically with a PHY capable of only a fraction of the data rate? Further...

Hi, I have python version 2.7 in Linux and now want to include the modules GSR(gadget snapshot reader) and pygadget ...can anyone please suggest what to proceed?? Thanks Apashanka

Dear Everybody, I don't know where to begin. So Here is the problem: $\newcommand{\Z}{\mathbb{Z}}$ Prove that if $[a]$ and $[b]$ are in ${\Z / n\Z}^{\times}$, then $[a] \times [b]$ is in ${\Z / n\Z}^{\times}$. Thanks, Cbarker1

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help in order to fully understand the proof of Proposition 4.3.14 ... ... Proposition 4.3.14 reads as follows: In the above proof by Bland we read...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need yet further help in order to fully understand the proof of Proposition 4.3.14 ... ... Proposition 4.3.14 reads as follows: In the above proof by...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need yet further help in order to fully understand the proof of Proposition 4.3.14 ... ... Proposition 4.3.14 reads as follows: In the above proof by...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some further help in order to fully understand the proof of Proposition 4.3.14 ... ... Proposition 4.3.14 reads as follows: In the above proof by...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help in order to fully understand the proof of Proposition 4.3.14 ... ... Proposition 4.3.14 reads as follows: In the above proof by Bland we...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some further help in order to fully understand the proof of Lemma 4.3.12 ... ... Lemma 4.3.12 reads as follows:My question is as follows: In the...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help in order to fully understand the proof of Proposition 4.3.12 ... ... Proposition 4.3.12 reads as follows:In the above proof by Bland we read...

Let's suppose that I have an element ##e## of order ##p## in the group of complex numbers whose elements all have order ##p^n## for some ##n\in\mathbb{N}## (henceforth called ##K##), and the module generated by ##(e)## is irreducible. How do I show that the injective hull of the module...

Problem. Let ##p## be a prime integer. Let ##Z_p^\infty## be the set of complex numbers having order ##p^n## for some ##n \in \mathbb{N}##, regarded as an abelian group under multiplication. Show that ##Z_p^\infty## has an unique simple submodule. Attempted solution. The collection of all...

Given: the short exact sequences 0 → M → E → K → 0 and 0 → M → E' → K' → 0 where M is a left R-module and E and E' are injective left R-modules. Prove: E ⊕ K' ≅ E' ⊕ K. First, let f be the morphism represented by M → E and g be the morphism represented by M → E'. Therefore we can construct a...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need some help to fully understand the proof of Lemma 4.3.10 ... ... Lemma 4.3.10 and its proof read as...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some further help to fully understand the proof of part of Proposition 4.2.16 (Jordan-Holder) ... ... Proposition 4.2.16 reads as follows...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.16 (Jordan-Holder) ... ... Proposition 4.2.16 reads as follows: Near the middle of the above...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.14 ... ... Proposition 4.2.14 reads as follows...

Let ##M## be a (right) R-module, and ##A## and ##B## two submodules of ##M##. If ##A = B ##, then we know that ##\frac{M}{A} = \frac{M}{B}##. But is the converse also true: If ##\frac{M}{A} = \frac{M}{B}##, can we conclude that ##A = B ## ? I doubt it, but I cannot find the answer. Maybe...

I am reading Dummit and Foote's book: "Abstract Algebra" (Third Edition) ... I am currently studying Chapter 10: Introduction to Module Theory ... ... I need some help with an aspect of Dummit and Foote's Section 10.3 Basic Generation of Modules, Direct Sums and Free Modules ... ... The...

I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.2 Free Modules ... ... I need help with some aspects of the proof of Proposition 2.2.6 ... Proposition 2.2.6 and its proof read as follows: Near the end of Bland's proof we read the following: "...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.11 ... ... Proposition 4.2.11 reads as follows:I need help with the Proof of $$(1)...

I have been reading about Rings and Modules. I am trying reconcile my understanding with Lie groups. Let G be a Matrix Lie group. The group acts on itself by left multiplication, i.e, Lgh = gh where g,h ∈ G Which corresponds to a translation by g. Is this an example of a module over a ring...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some further help to fully understand the proof of part of Proposition 4.2.10 ... ... Proposition 4.2.10 reads as follows:In the above proof by Bland we read the...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.10 ... ... Proposition 4.2.10 reads as follows:My questions are as follows:Question 1 In the...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some further help to fully understand the proof of part of Proposition 4.2.7 ... ... Proposition 4.2.7 reads as...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.7 ... ... Proposition 4.2.7 reads as follows:https://www.physicsforums.com/attachments/8208In...

Problem/Exercise $$M$$ is an R-module. $$N$$ is a submodule of M. $$N$$ and $$M/N$$ are Noetherian Show that $$M$$ is Noetherian ... ==================================== Progress so far ...Let $$K$$ be a submodule of $$M$$ ... must show $$K$$ is fingen ... Consider the mapping $$\pi \ : \...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Corollary 4.2.6 ... ... Corollary 4.2.6 reads as follows: Bland gives a statement of Corollary 4.2.6 but does...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.5 ... ... Proposition 4.2.5 reads as follows: https://www.physicsforums.com/attachments/8189My...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.5 ... ... Proposition 4.2.5 reads as follows: My questions are as follows:Question 1 In the...

I am reading T. S. Blyth's book: Module Theory: An Approach to Linear Algebra ... I am focused on Chapter 2: Submodules; intersections and sums ... and need help with the proof of Theorem 2.3 ... Theorem 2.3 reads as follows:In the above proof we read the following: " ... ... A linear...

I am reading T. S. Blyth's book: Module Theory: An Approach to Linear Algebra ... I am focused on Chapter 2: Submodules; intersections and sums ... and need help with the proof of Theorem 2.3 ... Theorem 2.3 reads as follows: In the above proof we read the following: " ... ... A linear...

In Chapter 1 of his book: "Modules and Rings", John Dauns (on page 7) considers a subset $$T$$ of an R-module $$M$$ and defines the R-submodule generated by $$T$$ ... for which he uses the notation $$\langle T \rangle$$ ... ... as follows:Now, note that Dauns (in Section 1-2.5) also defines...

In Chapter 1 of his book: "Modules and Rings", John Dauns (on page 7) considers a subset ##T## of an R-module ##M## and defines the R-submodule generated by ##T## ... for which he uses the notation ##\langle T \rangle## ... ... as follows: Now, note that Dauns (in Section 1-2.5) also defines...

Let ##M## be a left R-module and ##f:M \to M## an R-endomorphism. Consider this infinite descending sequence of submodules of ##M## ##M \supseteq f(M) \supseteq f^2(M) \supseteq f^3(M) \supseteq \cdots (1)## Can anybody show that the sequence (1) is strictly descending if ##f## is injective...

I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 4.1 Generating and Cogenerating Classes ... ... I need some help in order to make a meaningful start on Problem 2, Problem Set 4.1 ... Problem 2, Problem Set 4.1 reads as follows:( *** NOTE ...

I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ... I need help to make a meaningful start on Problem 14 of Problem Set 2.1 ... Problem 14 of Problem Set 2.1 reads as follows:I am somewhat overwhelmed by...

I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 4.2 Noetherian and Artinian Modules ... ... I need some help in order to make a meaningful start on Problem 1, Problem Set 4.1 ... Problem 1, Problem Set 4.1 reads as follows: Can someone...

Homework Statement I am reading Paul E. Bland's book "Rings and Their Modules" ... Currently I am focused on Section 4.2 Noetherian and Artinian Modules ... ... I need some help in order to make a meaningful start on Problem 1, Problem Set 4.1 ... Problem 1, Problem Set 4.1 reads as...