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

    MHB 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...
  2. Math Amateur

    Composition Series of Modules .... Remarks by Cohn

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series ... ) In Chapter 2: Linear Algebras and Artinian Rings, on Page 61 we find a definition of a refinement of a chain and a definition of a composition series. The relevant text on page 61 is as...
  3. Math Amateur

    Corollary to Correspondence Theorem for Modules

    I am reading Joseph J. Rotman's book: Advanced Modern Algebra and I am currently focused on Section 6.1 Modules ... I need some help with the proof of Corollary 6.25 ... Corollary to Theorem 6.22 (Correspondence Theorem) ... ... Corollary 6.25 and its proof read as follows: Can someone explain...
  4. Math Amateur

    Rotman's Remarks on Modules in Context of Chain Conditions

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  5. Math Amateur

    MHB How Does the Correspondence Theorem Prove Maximal Submodules and Simplicity?

    I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 7.1 Chain Conditions (for modules) ... I need some help in order to gain a full understanding of some remarks made in AMA on page 526 on modules in the context of chain conditions and...
  6. Math Amateur

    MHB Corollary to Correspondence Theorem for Modules

    I am reading Joseph J. Rotman's book: Advanced Modern Algebra and I am currently focused on Section 6.1 Modules ... I need some help with the proof of Corollary 6.25 ... Corollary to Theorem 6.22 (Correspondence Theorem) ... ... Corollary 6.25 and its proof read as follows:Can someone explain...
  7. Math Amateur

    MHB Correspondence Theorem for Modules - Rotman, Section 6.1

    I am reading Joseph J. Rotman's book: Advanced Modern Algebra and I am currently focused on Section 6.1 Modules ... I need some help with the proof of Theorem 6.22 (Correspondence Theorem) ... ... Theorem 6.22 and its proof read as...
  8. Math Amateur

    MHB Noetherian Rings and Modules: Theorem 2.2 - Cohn - Section 2.2 Chain Conditions

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  9. Math Amateur

    MHB Paul E Bland's "Direct Product of Modules" Definition - Category-Oriented

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  10. Math Amateur

    MHB Noetherian Modules - Maximal Condition - Berrick and Keating Ch. 3, page 111

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  11. Math Amateur

    MHB Theorem 3.1.4 - Berrick and Keating - Noetherian Rings and Modules

    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 help with the proof of Theorem 3.1.4. A brief explanation of the proof precedes the...
  12. Math Amateur

    MHB Corollary 3.1.3 - Berrick and Keating - Noetherian Modules

    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 help with the proof of Corollary 3.1.3. The statement of Corollary 3.1.3 reads as...
  13. caffeinemachine

    MHB Tensor Product of Two Finitely Generated Modules Over a Local Ring

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  14. K

    Bosonic Strings and their Verma modules

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  15. K

    MHB Proving that two quotient modules are isomorphic

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  16. Math Amateur

    MHB Switch from rings and modules to analysis

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  17. J

    Why Isn't My Thermoelectric Cooling Module Achieving Expected Performance?

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  18. Math Amateur

    MHB Noetherian Modules - Bland Proposition 4.2.3

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Chapter 4, Section 4.2 on Noetherian and Artinian Modules and need help with fully understanding Proposition 4.2.3, particularly assertion (2). Bland's statement of Proposition 4.2.3 reads as follows...
  19. Math Amateur

    MHB Noetherian Modules: ACC, Finite Ascending Chain Definition - Bland

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Chapter 4, Section 4.2 on Noetherian and Artinian modules and need help with the definition of a noetherian module - in particular I need help with the nature of an ascending chain of submodule ...
  20. Math Amateur

    MHB Noetherian Modules - Bland - Proposition 4.2.3 - (3) => (1)

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Chapter 4, Section 4.2 on Noetherian and Artinian modules and need help with the proof of (3) \Longrightarrow (1) in Proposition 4.2.3. Proposition 4.2.3 and its proof read as follows: The first...
  21. Math Amateur

    MHB Noetherian Modules - Bland - Proposition 4.2.3

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Chapter 4, Section 4.2 on Noetherian and Artinian modules and need help with the proof of (2) \Longrightarrow (3) in Proposition 4.2.3. Proposition 4.2.3 and its proof read as follows: In the proof...
  22. Math Amateur

    MHB Modules - Generating and Cogenerating Classes - Bland - Chapter 4, Section 4.1

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Chapter 4, Section 4.1 on generating and cogenerating classes and need help with the proof of Proposition 4.1.1. Proposition 4.1.1 and its proof read as follows: I need some help with what seems a...
  23. Math Amateur

    MHB Help with Paul E. Bland's Division Rings and IBN-rings Prop 2.2.10 Proof

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 2.2 on free modules and need help with the proof of Proposition 2.2.10. Proposition 2.2.10 and its proof read as follows:My question/problem is concerned with Bland's proof of Proposition 2.2.10...
  24. Math Amateur

    MHB Free Modules w/Multiple Bases: 2nd Issue w/Example 5 (Paul E. Bland)

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  25. Math Amateur

    MHB Free Modules With More Than One Basis - Bland - Example 5, page 56

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 2.2 on free modules and need help with Example 5 showing a module with two bases ... ... Example 5 reads as follows:I am having trouble understanding the notation and meaning of M = \bigoplus_{...
  26. Math Amateur

    Free Modules: Bland Corollary 2.2.4 - Issue on Finite Generation

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 2.2 on free modules and need help with the proof of Corollary 2.2.4. Corollary 2.2.4 and its proof read as follows: In the second last paragraph of Bland's proof above we read: " ... ... If...
  27. Math Amateur

    MHB Free Modules - Another issue regarding Bland Proposition 2.2.4

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 2.2 on free modules and need help with the proof of Corollary 2.2.4 - that is some further help ... (for my first problem with the proof see my post...
  28. Math Amateur

    MHB Free Modules: Solving Issue of Finite Generation Corollary 2.2.4

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 2.2 on free modules and need help with the proof of Corollary 2.2.4. Corollary 2.2.4 and its proof read as follows: In the second last paragraph of Bland's proof above we read: " ... ... If...
  29. Math Amateur

    MHB Free Modules - Another problem regarding Bland Proposition 2.2.3

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 2.2 on free modules and need help with the proof of Proposition 2.2.3 - that is some further help ... (for my first problem with the proof see my post...
  30. Math Amateur

    MHB Free Modules - Bland - Proposition 2.2.3

    I am reading Paul E. Bland's book, "Rings and Their Modules". I am trying to understand Section 2.2 on free modules and need help with the proof of Proposition 2.2.3. Proposition 2.2.3 and its proof read as follows:In the text above Bland writes: " ... ... The proof of the equivalence of (1)...
  31. Math Amateur

    MHB External and Internal Direct Sums - Bland - Rings and Their Modules

    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...
  32. Math Amateur

    MHB The Theory of Modules and Number Theory

    I have recently been doing some reading (skimming really) some books on number theory, particularly algebraic number theory. While number theory seems to draw heavily on rings and fields (especially some special types of rings like Euclidean rings and domains, unique factorization domains etc)...
  33. Math Amateur

    MHB Modules - Northott: Proposition 1

    In D. G. Northcott's book: Lessons on Rings, Modules and Multiplicities, Proposition 1 reads as follows:The first line of the above proof reads as follows: "Since 0_R + 0_R = 0_R, the definition of an R-module shows that 0_Rx = (0_R + 0_R)x = 0_Rx + 0_Rx, whence 0_Rx = 0_M, because M is a...
  34. M

    Semisimple rings with a unique maximal ideals

    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...
  35. Math Amateur

    MHB Direct Sums of Copies of Modules - B&K - Exercise 2.1.6 (iii)

    I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). In Chapter2: Direct Sums and Short Exact Sequences we find Exercise 2.1.6 part (iii). I need some help to get started on this exercise. Exercise 2.1.6 reads as follows: I am...
  36. Math Amateur

    MHB Ordered Index Sets for Direct Sums & Products of Modules: Explained by B&K

    I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K). In Chapter2: Direct Sums and Short Exact Sequences in Sections 2.1.14 and 2.1.15 B&K deal with ordered index sets in the context of direct sums and products of modules. Although it...
  37. Math Amateur

    MHB Free modules, M_n(R)-modules and M_N(R)-R-bimodules - 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). In Chapter 1: Basics under "1.2.3 Bimodules" B&K introduce some modules that they say play a key role in the text. To ensure I understood these I wrote out some details of each...
  38. Math Amateur

    MHB Theorem on the Lengths of Modules - Cohn, Theorem 2.5

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series)[/COLOR] In Chapter 2: Linear Algebras and Artinian Rings, on Page 61, Cohn presents Theorem 2,5 concerning the lengths of modules. Cohn indicates that the proof of this theorem is...
  39. Math Amateur

    MHB Modules of Finite Length - Cohn, page 61

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 2: Linear Algebras and Artinian Rings, on Page 61 we find a definition of the length of a module. Some analysis follows, as does a statement of Theorem 2.5. I need help to understand...
  40. Math Amateur

    MHB First Isomorphism Theorem for Modules - Cohn Theorem 1.17

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 1: Basics we find Theorem 1.17 (First Isomorphism Theorem for Modules) regarding module homomorphisms and quotient modules. I need help with some aspects of the proof. Theorem 1.17...
  41. Math Amateur

    MHB Quotient Modules and Module Homomorphisms - Cohn - Corollary 1.16

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 1: Basics we find Corollary 1.16 on module homomorphisms and quotient modules. I need help with some aspects of the proof. Corollary 1.16 reads as follows: In the above text...
  42. Math Amateur

    MHB Quotient Modules and Homomorphisms

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 1: Basics we find Theorem 1.15 on module homomorphisms and quotient modules. I need help with some aspects of the proof. Theorem 1.15 reads as follows: In the proof of the...
  43. Math Amateur

    MHB Modules in Cohn's book on ring theory - simple notational issue

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 1: Basics, on Page 33 we find a definition of a module homomorphism (or R-linear mapping) and a definition of Hom. I need help to interpret one of Cohn's expressions when he deals...
  44. Math Amateur

    MHB Chains of Modules and Composition Series

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series)[/COLOR] In Chapter 2: Linear Algebras and Artinian Rings, on Page 61 we find a definition of a refinement of a chain and a definition of a composition series. The relevant text on page 61 is as...
  45. Math Amateur

    MHB Finitely Generated Modules and Ascending Chains

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 2: Linear Algebras and Artinian Rings we find Proposition 2.4 on finitely generated modules and chain conditions. I need help with some aspects of the proof. Proposition 2.4 reads as...
  46. A

    How Does a Quotient Module Relate to Its Generators in a Local Ring?

    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...
  47. Math Amateur

    MHB Noetherian Modules and Finitely Generated Modules

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 2: Linear Algebras and Artinian Rings we find Theorem 2.2 on Noetherian modules. I need help with showing that "every module of M is finitely generated" implies that "M is Noetherian"...
  48. Math Amateur

    MHB Noetherian Modules - Cohn Theorem 2.2

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 2: Linear Algebras and Artinian Rings we find Theorem 2.2 on Noetherian modules. I need help with some aspects of the proof. Theorem 2 reads as follows: In the proof of (c)...
  49. NATURE.M

    Troubleshooting Xbee Module Data Transmission with Accelerometer Sensor

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  50. Math Amateur

    MHB Quick Question on Modules and Orthogonal Idempotents

    I am reading Berrick and Keating's book on Rings and Modules. Section 2.1.9 on Idempotents reads as follows: https://www.physicsforums.com/attachments/3097 https://www.physicsforums.com/attachments/3098So, on page 43 we read (see above) ... " ... ... Note that, conversely, a full set of...
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