Modules Definition and 224 Threads

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...

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...

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...

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...

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...

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...

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...

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 a feature of the Theorem...

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 am trying to fully understand Bland's definition of a direct product ... and to understand the motivation for the definition ... and the implications of...

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...

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...

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...

Problem. Let $R$ be a local ring (commutative with identity) ans $M$ and $N$ be finitely generated $R$-modules. If $M\otimes_R N=0$, then $M=0$ or $N=0$. The problem clearly seems to be an application of the Nakayama lemma. If we can show that $M=\mathfrak mM$ or $N=\mathfrak mN$, where...

Hi there! I have som troubles with representation theory. It is obvious that bosonic strings fields $X^{\mu}$ has zero conformal dimension $h=0$. But when one builds Verma module (open string for example) highest weight state has the following definition $$ L_0 \vert h \rangle = 1 \vert h...

Given a ring $R$ and $R$-modules $A,B,C,D$ such that $$\sigma:A \rightarrow B, \tau: C \rightarrow D, \rho: A \rightarrow C, \kappa: B \rightarrow D, \ \mathrm{and} \ \kappa \circ \sigma = \tau \circ \rho,$$ where $\sigma, \tau, \rho, \kappa$ are homomorphisms and $\rho, \kappa$ are...

I am temporarily switching my studies from abstract algebra to mathematical analysis. I am thinking of reading the following book: Principles of Mathematical Analysis by Walter Rudin. What books to MHB members advise me to use in order to gain a full understanding of undergraduate level...

hI'm hoping someone here is well versed at using thermoelectric cooling modules. I'm not able achieve anywhere near the cooling capacity shown in the datasheets and I'm pretty sure I've covered all possible things that could be going wrong. All I'm trying to do is maintain chilled water...

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...

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 ...

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...

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...

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...

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...

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 ... ... Thanks to Caffeinemachine, I have largely clarified one issue/problem I had with Example 5, but now have a...

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_{...

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...

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...

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...

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...

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)...

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...

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)...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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"...

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)...

So quick question: Basically I have my xbee pro 900 configured to an Arduino to read data from an accelerometer. The data will then get displayed in the xctu console on my computer transmitted wirelessly of course. Now I did this same procedure with a temp. sensor and the data was displayed...

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...