Modules Definition and 224 Threads

Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a system into varying degrees of interdependence and independence across and "hide the complexity of each part behind an abstraction and interface". However, the concept of modularity can be extended to multiple disciplines, each with their own nuances. Despite these nuances, consistent themes concerning modular systems can be identified.

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  1. jack action

    Fixing Linux kernel not found

    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...
  2. Euge

    POTW Flat Modules and Intersection

    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##.
  3. berkeman

    How are blood glucose monitoring modules installed in arms?

    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...
  4. Eclair_de_XII

    Python Should I rewrite my modules in order to implement json/pickle?

    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...
  5. Arman777

    Python Importing multiple modules from a subfolder to another folder

    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...
  6. F

    Python Understanding where pip saves the downloaded Python modules

    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...
  7. Eclair_de_XII

    Python Writing modules that require supplementary files

    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...
  8. F

    Python Understanding Python Module Installation with Pip

    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...
  9. F

    How are 2D electrostatic modules approximated in COMSOL?

    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?
  10. trurle

    AC-DC Power Supply Modules: Non-Isolated Topology

    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...
  11. A

    Anyone familiar with SFP Fiber Optic Modules?

    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...
  12. A

    Python For the modules GSR (gadget snapshot reader) and pygadget

    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
  13. C

    MHB Can $[a] \times [b]$ be an element of ${\Z / n\Z}^{\times}$?

    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
  14. Math Amateur

    I Primitive Elements and Free Modules .... ....

    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...
  15. Math Amateur

    MHB Understanding Bland's Proof of Proposition 4.3.14: Primitive Elements of Modules

    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...
  16. Math Amateur

    MHB Exploring Paul E. Bland's "Rings and Their Modules" - Proposition 4.3.14

    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...
  17. Math Amateur

    MHB Help Understanding Bland's Proposition 4.3.14 in Rings and Their Modules

    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...
  18. Math Amateur

    MHB Primitive Elements and Free Modules .... ....

    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...
  19. Math Amateur

    MHB Understanding Lemma 4.3.12 in Paul E. Bland's Book: "Rings and Their Modules"

    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...
  20. Math Amateur

    MHB Why Does d and 1 Being Associates Imply 1 is a GCD?

    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...
  21. T

    A Is the Injective Hull of an Irreducible Module in Group K?

    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...
  22. T

    A Construct a unique simple submodule

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  23. T

    A Proving the Dual of Schanuel's Lemma

    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...
  24. Math Amateur

    MHB Why Is the Containment xR ⊆ x1R Proper in Lemma 4.3.10?

    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...
  25. Math Amateur

    MHB Jordan-Holder Theorem for Modules .... .... Another Two Questions ....

    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...
  26. Math Amateur

    MHB Jordan-Holder Theorem for Modules .... ....

    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...
  27. Math Amateur

    MHB Composition Series and Noetherian and Artinian Modules ....

    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...
  28. S

    I A little problem about quotient modules

    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...
  29. Math Amateur

    MHB Generation of Modules .... Dummit and Foote, Section 10.3 .... ....

    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...
  30. Math Amateur

    MHB Homomorphic Images of Free Modules .... Bland, Proposition 2.2.6 .... ....

    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: "...
  31. Math Amateur

    MHB How Does Corollary 4.2.6 Prove M is Noetherian?

    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)...
  32. N

    I Rings, Modules and the Lie Bracket

    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...
  33. Math Amateur

    MHB Understanding Bland's Proposition 4.2.10 in Rings and Modules

    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...
  34. Math Amateur

    MHB Finite Sum of Indecomposable Modules .... Bland, Proposition 4.2.10 .... ....

    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...
  35. Math Amateur

    MHB Noetherian Modules: Direct Sums & Bland Proposition 4.2.7

    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...
  36. Math Amateur

    MHB Direct Sums of Noetherian Modules .... Bland Proposition 4.2.7 .... ....

    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...
  37. Math Amateur

    MHB Noetherian Modules, Submodules and Factor Modules .... Problem/Exercise

    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 \ : \ M \to M/N where \pi (x) =...
  38. Math Amateur

    MHB Noetherian Modules and Short Exact Sequences .... Bland, Corollary 4.2.6 ....

    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...
  39. Math Amateur

    MHB Can Submodules of a Noetherian Module Fail to Intersect with a Given Submodule?

    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...
  40. Math Amateur

    I Submodules and Factor Modules of a Noetherian Module ....

    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...
  41. Math Amateur

    MHB Generating/spanning modules and submodules .... .... Blyth Theorem 2.3

    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...
  42. Math Amateur

    I Generating modules and sub modules Blyth Theorem 2.3

    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...
  43. Math Amateur

    MHB Generating/spanning modules and submodules .... ....

    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 \sum M_i = \langle...
  44. Math Amateur

    I Generating/spanning modules and submodules .... ....

    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...
  45. S

    A A decreasing sequence of images of an endomorphisme

    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...
  46. Math Amateur

    MHB Modules Generated by Sets of Submodules .... .... Bland Problem 2, Problem Set 4.1 .... ....

    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 ...
  47. Math Amateur

    MHB Direct Sums and Factor Modules .... Bland Problem 14, Problem Set 2.1

    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...
  48. Math Amateur

    MHB Start on Bland Problem 1, Problem Set 4.1: Generating & Cogenerating Modules

    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...
  49. Math Amateur

    Modules Generated by Sets of Submodules .... ....

    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...
  50. Math Amateur

    MHB Noetherian Modules - Bland, Example 3, Section 4.2 ....

    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 fully understand Example 3, Section 4.2 ... Example 3, Section 4.2 reads as follows: My questions are as...
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