What is Modules: Definition and 224 Discussions

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. Math Amateur

    I 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...
  2. S

    A On a finitely generated submodule of a direct sum of modules....

    I am new on this forum, this is my gift for you. Suppose ##(M_i)_{i \in I}## is a family of left ##R##-modules and ##M = \bigoplus_{i \in I} M_i## (external direct sum). Suppose ##N = \langle x_1, \cdots ,x_m \rangle## is a finitely generated submodule of ##M##. Then for each ##j = 1, \cdots...
  3. Math Amateur

    MHB Finitely Generated Modules - Bland Problem 1(a), Problem Set 2.2

    I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.2 Free Modules ... ... I need someone to check my solution to the first part of Problem 1(a) of Problem Set 2.2 ... Problem 1(a) of Problem Set 2.2 reads as follows: My solution/proof of the...
  4. Math Amateur

    Finitely Generated Modules - Bland Problem 1(a), Set 2.2

    Homework Statement I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.2 Free Modules ... ... I need someone to check my solution to the first part of Problem 1(a) of Problem Set 2.2 ... Problem 1(a) of Problem Set 2.2 reads as follows: Can...
  5. Math Amateur

    MHB How Is u_alpha an Injective Mapping in Proposition 2.1.4?

    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 with some aspects of the proof of Proposition 2.1.4 ... Proposition 2.1.4 and its proof read as follows: In the above proof by Paul Bland we...
  6. Math Amateur

    I Direct Products of Modules .... Canonical Injections ....

    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 with some aspects of the proof of Proposition 2.1.4 ... Proposition 2.1.4 and its proof read as follows: In the above proof by Paul Bland we...
  7. Math Amateur

    MHB Rings and Their Modules .... ....

    For those MHB members who are interested in the theory of rings and their modules at the advanced undergraduate and beginning graduate levels it is hard to go past the book by Paul E. Bland and the book by Berrick and Keating ... namely ... "Rings and Their Modules" by Paul E. Bland (De Gruyter...
  8. Math Amateur

    MHB Understanding Proposition 2.1.1 in Paul E. Bland's Rings & Modules

    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 with another aspect of the proof of Proposition 2.1.1 ... Proposition 2.1.1 and its proof read as follows...
  9. Math Amateur

    I Direct Products of Modules ....Another Question .... ....

    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 with another aspect of the proof of Proposition 2.1.1 ... Proposition 2.1.1 and its proof read as follows:In the above proof by Paul Bland we...
  10. Math Amateur

    MHB Direct Products of Modules .... Bland Proposition 2.1.1 .... ....

    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 with some aspects of the proof of Proposition 2.1.1 ... Proposition 2.1.1 and its proof read as follows: In the statement of the above...
  11. Math Amateur

    I Direct Products of Modules .... Bland Proposition 2.1.1 ....

    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 with some aspects of the proof of Proposition 2.1.1 ... Proposition 2.1.1 and its proof read as follows: In the statement of the above...
  12. Math Amateur

    MHB Noetherian Modules .... Cohn Theorem 2.2 .... ....

    I am reading P.M. Cohn's book: Introduction to Ring Theory (Springer Undergraduate Mathematics Series) ... ... I am currently focused on Section 2.2: Chain Conditions ... which deals with Artinian and Noetherian rings and modules ... ... I need help with understanding an aspect of the proof of...
  13. Math Amateur

    I Noetherian Modules .... Cohn Theorem 2.2 .... ....

    I am reading P.M. Cohn's book: Introduction to Ring Theory (Springer Undergraduate Mathematics Series) ... ... I am currently focused on Section 2.2: Chain Conditions ... which deals with Artinian and Noetherian rings and modules ... ... I need help with understanding an aspect of the proof of...
  14. FallenApple

    I Advantages of Vector Spaces over Modules

    I know that vector spaces have more structure as they are defined over fields and that modules are defined over rings. But it's hard to think of a situation where a using a ring clearly backfires. Is it just because a ring doesn't have an inverse for the second operation? For a module over Z...
  15. R

    Other A module choice between Cosmology and Fluid Mechanics....

    Hi all, I study Mathematics and Theoretical Physics at university. Next year, I'll have a choice between three modules; Thermodynamics and Statistical Physics Cosmology Fluid Mechanics My main interest is Condensed Matter Physics; therefore, thermodynamics and statistical physics is a given...
  16. Math Amateur

    History of modules in mathematics ....

    I would like to read a book, the chapter of a book or a website article (preferably comprehensive) on the history of the concept of a module and the development of the theory of modules in mathematics ... Can anyone help with some good references ... Peter
  17. Math Amateur

    History History of the theory of modules ....

    Does anyone know of a book or web page that gives a history of the concept of a module and the history of modules in algebra ... i have not not been able to find a book that covers the history of modules nor a website ... Help will be appreciated... Peter
  18. Math Amateur

    MHB Exploring Proposition 6.1.7 and its Proof in Bland's "Rings and Their Modules"

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 6.1 The Jacobson Radical ... ... I need help with the proof of Proposition 6.1.7 ... Proposition 6.1.7 and its proof read as follows: In the above text from Bland, in the proof of (1), we read the...
  19. Math Amateur

    MHB Simple Modules and Right Annihilator Ideals .... Bland, Proposition 6.1.7 ....

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 6.1 The Jacobson Radical ... ... I need help with the proof of Proposition 6.1.7 ... Proposition 6.1.7 and its proof read as follows: In the above text from Bland ... in the proof of (1) we read the...
  20. Math Amateur

    I Simple Modules and Right Annihilator Ideals ....

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 6.1 The Jacobson Radical ... ... I need help with the proof of Proposition 6.1.7 ...Proposition 6.1.7 and its proof read as follows: In the above text from Bland ... in the proof of (1) we read the...
  21. Math Amateur

    MHB Simple Modules and Maximal Right Ideals .... Bland, Proposition 6.1.7 ....

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 6.1 The Jacobson Radical ... ... I need help with the proof of Proposition 6.1.7 ... Proposition 6.1.7 and its proof read as follows:https://www.physicsforums.com/attachments/6388...
  22. Math Amateur

    I Simple Modules and Maximal Right Ideals ....

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 6.1 The Jacobson Radical ... ... I need help with the proof of Proposition 6.1.7 ...Proposition 6.1.7 and its proof read as follows: In the above text from Bland, in the proof of (1) we read the following...
  23. Math Amateur

    I Simple Modules and quotients of maximal modules, Bland Ex 13

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Chapter 1, Section 1.4 Modules ... ... I need help with the proving a statement Bland makes in Example 13 ... ... Example 13 reads as follows: In the above text from Bland, we read the following: " ... If ##N## is...
  24. Math Amateur

    MHB Proof of Bland's Example 13: Simple Modules and Quotients of Maximal Modules

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Chapter 1, Section 1.4 Modules ... ... I need help with the proving a statement Bland makes in Example 13 ... ... Example 13 reads as follows:In the above text from Bland, we read the following: " ... If N is a...
  25. Math Amateur

    MHB Finitely Generated Modules and Maximal Submodules

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 6.1 The Jacobson Radical ... ... I need help with an aspect of Proposition 6.1.2 ... ... Proposition 6.1.2 relies on Zorn's Lemma and the notion of inductive sets ... ... so I am providing a short note...
  26. Math Amateur

    I Finitely Generated Modules and Maximal Submodules

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 6.1 The Jacobson Radical ... ... I need help with an aspect of Proposition 6.1.2 ... ... Proposition 6.1.2 relies on Zorn's Lemma and the notion of inductive sets ... ... so I am providing a short note from...
  27. Math Amateur

    MHB Artinian Modules - Bland - Proposition 4.24

    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 4.2.4 ... ... Proposition 4.2.4 reads as follows: I need help to fully understand Part of the...
  28. Math Amateur

    I Artinian Modules - Bland - Proposition 4.24

    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 4.2.4 ... ... Proposition 4.2.4 reads as follows: I need help to fully understand Part of the...
  29. Math Amateur

    I Noetherian Modules and Submodules

    I am reading J A Beachy's Book, Introductory Lectures on Rings and Modules"... ... I am currently focused on Chapter 2: Modules ... and in particular Section 2.4: Chain Conditions ... I need help with the proof of Proposition 2.4.5 ... Proposition 2.4.5 reads as follows: In the above text...
  30. Math Amateur

    MHB Noetherian Modules and Submodules - J A Beachy, Proposition 2.4.5 .... ....

    I am reading J A Beachy's Book, Introductory Lectures on Rings and Modules"... ... I am currently focused on Chapter 2: Modules ... and in particular Section 2.4: Chain Conditions ... I need help with the proof of Proposition 2.4.5 ...Proposition 2.4.5 reads as follows...
  31. Math Amateur

    I Finitely Generated Modules .... another question

    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 the proof of Lemma 1.2.21 ... Lemma 1.2.21 and its proof reads as follows: In the above text (in the proof) by Berrick and Keating we read the following:"... ...
  32. Math Amateur

    MHB Finitely Generated Modules and Their Submodules .... Berrick and Keating ....

    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 the proof of Lemma 1.2.21 ... Lemma 1.2.21 and its proof reads as follows: Question 1In the above text by Berrick and Keating, we read the following:"... ... Since...
  33. Math Amateur

    I Finitely Generated Modules and Their Submodules ....

    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 the proof of Lemma 1.2.21 ... Lemma 1.2.21 and its proof reads as follows: Question 1In the above text by Berrick and Keating, we read the following:"... ...
  34. Math Amateur

    MHB Finitely Cogenerated Modules - Bland Definition 4.1.3, Page 105

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 4.1 on Generating and Cogenerating Classes ... ... In particular I am trying to understand and get a good sense of the notion of a module being finitely cogenerated ... Bland defines a finitely...
  35. S

    MHB Bland - Rings and their Modules 2011

    If the moderators agree, I have made a thread on the book of Bland - Rings and their Modules 2011.
  36. Math Amateur

    I Tensor Products of Modules - Bland - Remark, Page 65

    I am reading Paul E. Bland's book "Rings and Their Modules ... Currently I am focused on Section 2.3 Tensor Products of Modules ... ... I need some help in order to fully understand the Remark that Bland makes on Pages 65- 66 Bland's remark reads as follows: Question 1 In the above text by...
  37. Math Amateur

    MHB Tensor Products of Modules - Bland - Remark, Page 65

    I am reading Paul E. Bland's book "Rings and Their Modules ... Currently I am focused on Section 2.3 Tensor Products of Modules ... ... I need some help in order to fully understand the Remark that Bland makes on Pages 65- 66 Bland's remark reads as follows: Question 1 In the above text by...
  38. Math Amateur

    MHB Understanding Bland's Example: Free Modules & Directly Finite/Infinite R-Modules

    I am reading Paul E. Bland's book "Rings and Their Modules ... Currently I am focused on Section 2.2 Free Modules ... ... I need some help in order to fully understand Bland's Example on page 56 concerning directly finite and directly infinite R-modules ... ... Bland's Example on page 56...
  39. Math Amateur

    I Free Modules, Bases and Direct Sums/Products

    I am reading Paul E. Bland's book "Rings and Their Modules ... Currently I am focused on Section 2.2 Free Modules ... ... I need some help in order to fully understand the proof of the equivalence of (1) and (3) in Proposition 2.2.3 ... Proposition 2.2.3 and its proof reads as follows: Bland...
  40. Math Amateur

    MHB Free Modules, Bases and Direct Sums/Products - Bland, Proposition 2.2.3

    I am reading Paul E. Bland's book "Rings and Their Modules ... Currently I am focused on Section 2.2 Free Modules ... ... I need some help in order to fully understand the proof of the equivalence of (1) and (3) in Proposition 2.2.3 ... Proposition 2.2.3 and its proof reads as follows:Bland...
  41. M

    MHB Wondering if $R$ is a P.I.D.: Free Modules and Ideals

    Hey! :o Let $R$ be a commutative ring with unit. I want to show that if each $R$-submodule of a free $R$-module is free then $R$ is P.I.D.. From the thread http://mathhelpboards.com/linear-abstract-algebra-14/how-can-we-conclude-i-principal-ideal-18593.html we have that if an ideal of $R$ is...
  42. S

    Limiting factors of PV Modules and Thermal Collectors

    What limits the efficiency of a PV Module and a Thermal Collector? PV efficiecy is limited more by temperature in summer than by illumination in winter. Thermal Collector efficiency is limited by temperature and illumination in winter. The lower the temperature a thermal collector will be...
  43. Math Amateur

    MHB Tensor Products of Modules and Free Abelian Groups based on Cartesian Product

    I am reading Donald S. Passmore's book "A Course in Ring Theory" ... I am currently focussed on Chapter 9 Tensor Products ... ... I need help in order to get a full understanding of the free abelian group involved in the construction of the tensor product ... ... The text by Passmore...
  44. DiracPool

    Python Problems installing Python modules

    I've installed Python 3.5.1 successfully, and I've downloaded add-on modules such as numpy, scipy, matplotlib, etc, but when I try to import them into python I get the following: "ImportError: No module named 'numpy'", etc. There's no executable file in the numpy download to activate anything...
  45. Math Amateur

    MHB Extension of Scalars: Dummit & Foote's Section 10.4 Q&A

    I am reading Dummit and Foote's book: Abstract Algebra ... ... and am currently focused on Section 10.4 Tensor Products of Modules ... ... I have a basic question regarding the extension of the scalars ... Dummit and Foote's exposition regarding extension of the scalars reads as...
  46. O

    Thermoelectric Modules: Heat/Cold Ratio for Voltage

    What is the heat/cold ratio in thermoelectric modules to generate a specific voltage pls? old Bozo
  47. Math Amateur

    MHB C[T] modules - Berrick and Keating Exercise 1.2.8 (c)

    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 1.2.8 (c) ... Exercise 1.2.8 (c) reads as follows:Now ... from this exercise we have that M \equiv \mathbb{C}^3 ... and ... A = \begin{pmatrix} 0&1&1 \\...
  48. Math Amateur

    MHB Polynomials Acting on Spaces - B&K Ex. 1.2.2 (iv): An Intro by Peter

    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 in order to fully understand Example 1.2.2 (iv) [page 16] ... indeed, I am somewhat overwhelmed by this construction ... ... Example 1.2.2 (iv) reads as follows:My...
  49. Math Amateur

    Thread: Artinian Modules - Cohn Exercise 3, Section 2.2, page 65

    I am reading P.M. Cohn's book: Introduction to Ring Theory (Springer Undergraduate Mathematics Series) ... ... I am currently focused on Section 2.2: Chain Conditions ... which deals with Artinian and Noetherian rings and modules ... ... I need help to get started on Exercise 3, Section 2.2...
  50. Math Amateur

    Noetherian Modules - Maximal Condition - Berrick and Keating

    I am reading the book "An Introduction to Rings and Modules with K-theory in View" by A.J. Berrick and M.E. Keating ... ... I am currently focused on Chapter 3; Noetherian Rings and Polynomial Rings. I need someone to help me to fully understand the maximal condition for modules and its...
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