What is Rings: Definition and 437 Discussions

Fighting Network Rings, trademarked as RINGS, is a Japanese combat sport promotion that has lived three distinct periods: puroresu promotion from its inauguration to 1995, mixed martial arts promotion from 1995 to its 2002 disestablishment, and the revived mixed martial arts promotion from 2008 onward.
RINGS was founded by Akira Maeda on May 11, 1991, following the dissolution of Newborn UWF. At that time, Maeda and Mitsuya Nagai were the only two people to transfer from UWF, wrestlers such as Kiyoshi Tamura, Hiromitsu Kanehara and Kenichi Yamamoto would later also transfer from UWF International.

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  1. Math Amateur

    MHB Help with Problem 2(a), Rings & Modules, Paul E. Bland

    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 Problem 2(a) of Problem Set 2.1 ... Problem 2(a) of Problem Set 2.1 reads as follows: I am unsure of my solution to problem 2(a) and need...
  2. Math Amateur

    Direct Products of Rings and Ideals .... Bland Problem 2(a)

    Homework Statement 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 Problem 2(a) of Problem Set 2.1 ... Problem 2(a) of Problem Set 2.1 reads as follows: I am unsure of my solution to...
  3. 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...
  4. 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...
  5. Math Amateur

    MHB Third Isomorphism Theorem for Rings .... Bland Theorem 3.3.16 .... ....

    I am reading "The Basics of Abstract Algebra" by Paul E. Bland ... ... I am currently focused on Chapter 3: Sets with Two Binary Operations: Rings ... ... I need help with Bland's proof of the Third Isomorphism Theorem for rings ... Bland's Third Isomorphism Theorem for rings and its proof...
  6. Math Amateur

    I Third Isomorphism Theorem for Rings .... Bland Theorem 3.3.16

    I am reading "The Basics of Abstract Algebra" by Paul E. Bland ... ... I am currently focused on Chapter 3: Sets with Two Binary Operations: Rings ... ... I need help with Bland's proof of the Third Isomorphism Theorem for rings ... Bland's Third Isomorphism Theorem for rings and its proof...
  7. Math Amateur

    MHB Second Isomorphism Theorem for Rings .... Bland Theorem 3.3.15 .... ....

    I am reading "The Basics of Abstract Algebra" by Paul E. Bland ... ... I am currently focused on Chapter 3: Sets with Two Binary Operations: Rings ... ... I need help with Bland's proof of the Second Isomorphism Theorem for rings ... Bland's Second Isomorphism Theorem for rings and its proof...
  8. Math Amateur

    I Second Isomorphism Theorem for Rings .... Bland Theorem 3.3.1

    I am reading "The Basics of Abstract Algebra" by Paul E. Bland ... ... I am currently focused on Chapter 3: Sets with Two Binary Operations: Rings ... ... I need help with Bland's proof of the Second Isomorphism Theorem for rings ... Bland's Second Isomorphism Theorem for rings and its proof...
  9. Math Amateur

    MHB Quotient Rings .... Remarks by Adkins and Weintraub ....

    I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ... I am currently focused on Chapter 2: Rings ... I need help with fully understanding some remarks by Adkins and Weintraub on quotient rings on page 59 in Chapter 2 ... The remarks by Adkins...
  10. Math Amateur

    I Quotient Rings .... Remarks by Adkins and Weintraub ....

    I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ... I am currently focused on Chapter 2: Rings ... I need help with fully understanding some remarks by Adkins and Weintraub on quotient rings on page 59 in Chapter 2 ... The remarks by Adkins...
  11. Math Amateur

    MHB Division Rings and RIng Homomorphisms .... A&W Corollary 2.4 ....

    I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ... I am currently focused on Chapter 2: Rings ... I need help with an aspect of the proof of Corollary 2.4 ... ... Corollary 2.4 and its proof read as follows: In the above proof of Corollary...
  12. Math Amateur

    I Division Rings & Ring Homomorphisms .... A&W Corollary 2.4 ...

    I am reading "Algebra: An Approach via Module Theory" by William A. Adkins and Steven H. Weintraub ... I am currently focused on Chapter 2: Rings ... I need help with an aspect of the proof of Corollary 2.4 ... ... Corollary 2.4 and its proof read as follows: In the above proof of...
  13. C

    I Splitting ring of polynomials - why is this result unfindable?

    Assume that ##P## is a polynomial over a commutative ring ##R##. Then there exists a ring ##\tilde R## extending ##R## where ##P## splits into linear factor (not necessarily uniquely). This theorem, whose proof is given below, is difficult to find in the literature (if someone know a source, it...
  14. A

    Reflected rings from the chair leg

    This is a pic of sun rays shining on the metallic leg of the chair. The reflection comes on the floor of my room as rings. The source is just a square window. No other light sources in the room aside from reflection from the walls. The leg of the chair is cylinder in shape with smooth surface...
  15. A

    Newton's Rings and Thin Film Interference

    Homework Statement A total of 31 bright and 31 dark Newton's rings (not counting the dark spot at the center) are observed when 550-nm light falls normally on a planoconvex lens resting on a flat glass surface. How much thicker is the center than the edges? Homework Equations 2t = mλ 2t =...
  16. B

    Nontrivial Finite Rings with No Zero Divisors

    Homework Statement A finite ring with more than one element and no zero divisors is a division ring (Special case: a finite integral domain is a field) Homework EquationsThe Attempt at a Solution Let ##r \in R \setminus \{0\}##, and define ##f : R \setminus \{0\} \to R \setminus \{0\}## by...
  17. R

    Finding electric potential of two concentric rings

    Homework Statement Two concentric spheres have radii a and b with b>a. The region between them is filled with charge of constant density p. The charge density is zero everywhere else. Hence, find the electric field of all points , then find the electric potential. 2. Homework Equations [/B]The...
  18. Laurie K

    I Calculating the Optical Appearance of Rolling Rings: A Methodology by Ø. Grøn

    This post is about the methodology required to produce the Fig. 9 Part C “optical appearance” plot as calculated by Øyvind Grøn in "Space geometry in rotating reference frames: A historical appraisal". http://areeweb.polito.it/ricerca/relgrav/solciclos/gron_d.pdf In 2014 I contributed to a...
  19. Mr Davis 97

    Show whether two quotient rings are isomorphic

    Homework Statement Sorry for the multiple postings. I actually solved the other problems, so I have this last one: Are ##\mathbb{Z}_3[x] / \langle x^2 + 2x + 1 \rangle ## and ##\mathbb{Z}_3[x] / \langle x^2 + x + 2 \rangle## isomorphic Homework EquationsThe Attempt at a Solution It seems...
  20. Mr Davis 97

    Show that two rings are not isomorphic

    Homework Statement Show that the rings ##2 \mathbb{Z}## and ##3 \mathbb{Z}## are not isomorphic. Homework EquationsThe Attempt at a Solution I know how to show that two structures are isomorphic: find an isomorphism. However, I am not quite sure how to show that there exists no isomorphism at all
  21. N

    I Do quantum linked rings (quantum knots) have energy?

    How can we harness that energy? Is there any engine, heater, battery or other thing that uses it?
  22. Math Amateur

    MHB Rings of Fractions .... Lovett, Section 6.2, Proposition 6.2.6 .... ....

    I am reading Stephen Lovett's book, "Abstract Algebra: Structures and Applications" and am currently focused on Section 6.2: Rings of Fractions ... I need some help with the proof of Proposition 6.2.6 ... ... ... Proposition 6.2.6 and its proof read as follows: In the above proof by...
  23. Math Amateur

    I Rings of Fractions .... Lovett, Section 6.2, Proposition 6.2

    I am reading Stephen Lovett's book, "Abstract Algebra: Structures and Applications" and am currently focused on Section 6.2: Rings of Fractions ... I need some help with the proof of Proposition 6.2.6 ... ... ... Proposition 6.2.6 and its proof read as follows: In the above proof by Lovett...
  24. Math Amateur

    MHB Rings of Fractions .... Lovett, Section 6.2 ....

    I am reading Stephen Lovett's book, "Abstract Algebra: Structures and Applications" and am currently focused on Section 6.2: Rings of Fractions ... I need some help with some remarks following Definition 6.2.4 ... ... ... The remarks following Definition 6.2.4 reads as...
  25. Math Amateur

    I Rings of Fractions .... Lovett, Section 6.2 ....

    I am reading Stephen Lovett's book, "Abstract Algebra: Structures and Applications" and am currently focused on Section 6.2: Rings of Fractions ... I need some help with some remarks following Definition 6.2.4 ... ... ... The remarks following Definition 6.2.4 reads as follows: In the...
  26. Math Amateur

    MHB Rings Generated by Elements - Lovett, Example 5.2.1 .... ....

    I am reading Stephen Lovett's book, "Abstract Algebra: Structures and Applications" and am currently focused on Chapter 5 ... I need some help with Example 5.2.1 in Section 5.2: Rings Generated by Elements ... In the Introduction to Section 5.2.1 (see text above) Lovett writes: " ... ...
  27. Math Amateur

    I Rings Generated by Elements - Lovett, Example 5.2.1 .... ....

    I am reading Stephen Lovett's book, "Abstract Algebra: Structures and Applications" and am currently focused on Chapter 5 ... I need some help with Example 5.2.1 in Section 5.2: Rings Generated by Elements ... In the Introduction to Section 5.2.1 (see text above) Lovett writes: " ... ... R[S]...
  28. Math Amateur

    MHB Simple Unitial Rings .... centre is a field .... ? ....

    I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ... I need help with some remarks of Bresar in Example 1.21 on simple unital rings ... Example 1.21 reads as follows: In the...
  29. Math Amateur

    I Simple Unitial Rings .... centre is a field .... ? ....

    I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ... I need help with some remarks of Bresar in Example 1.21 on simple unital rings ... Example 1.21 reads as follows: In the above text...
  30. Math Amateur

    I Introduction to Simple Matrix Rings in Noncommutative Algebra

    I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ... I need help with some aspects of Bresar's Example 1.10 on a simple matrix ring over a division ring ... Example 1.10, including some...
  31. Math Amateur

    MHB Understanding Bresar's Example 1.10 on Simple Matrix Rings: Can Anyone Help?

    I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ... I need help with some aspects of Bresar's Example 1.10 on a simple matrix ring over a division ring ... Example 1.10...
  32. Math Amateur

    MHB Matrix Rings - Basic Problem with Meaning of Notation .... Rowen, Section 1.1 ....

    I am reading Louis Rowen's book, "Ring Theory"(Student Edition) ... I have a problem interpreting Rowen's notation in Section 1.1 Matrix Rings and Idempotents ... The relevant section of Rowen's text reads as follows: In the above text from Rowen, we read the following: " ... ... We obtain a...
  33. Math Amateur

    I Matrix Rings - Basic Problem with Meaning of Notation

    I am reading Louis Rowen's book, "Ring Theory" (Student Edition) ... I have a problem interpreting Rowen's notation in Section 1.1 Matrix Rings and Idempotents ... The relevant section of Rowen's text reads as follows: In the above text from Rowen, we read the following: " ... ... We obtain...
  34. Math Amateur

    MHB Finitely Generated Ideals and Noetherian Rings

    I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 5.3 Noetherian Rings ...I need some help with understanding the proof of Proposition 5.33 ... ...Proposition 5.33 reads as follows: https://www.physicsforums.com/attachments/6009 In the...
  35. Math Amateur

    I Finitely Generated Ideals and Noetherian Rings

    I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 5.3 Noetherian Rings ...I need some help with understanding the proof of Proposition 5.33 ... ...Proposition 5.33 reads as follows: In the above text from Rotman, in the proof of (ii)...
  36. Math Amateur

    MHB Triangular Matrix RIngs .... Lam, Proposition 1.17

    I am reading T. Y. Lam's book, "A First Course in Noncommutative Rings" (Second Edition) and am currently focussed on Section 1:Basic Terminology and Examples ... I need help with Part (1) of Proposition 1.17 ... ... Proposition 1.17 (together with related material from Example 1.14 reads...
  37. Math Amateur

    I Triangular Matrix RIngs .... Lam, Proposition 1.17

    I am reading T. Y. Lam's book, "A First Course in Noncommutative Rings" (Second Edition) and am currently focussed on Section 1:Basic Terminology and Examples ... I need help with Part (1) of Proposition 1.17 ... ... Proposition 1.17 (together with related material from Example 1.14 reads as...
  38. Math Amateur

    MHB Triangular Matrix Rings .... Another Question

    I am reading T. Y. Lam's book, "A First Course in Noncommutative Rings" (Second Edition) and am currently focussed on Section 1:Basic Terminology and Examples ... I need help with yet another aspect of Example 1.14 ... ... Example 1.14 reads as follows: Near the end of the above text from...
  39. Math Amateur

    I Triangular Matrix RIngs .... Another Question

    I am reading T. Y. Lam's book, "A First Course in Noncommutative Rings" (Second Edition) and am currently focussed on Section 1:Basic Terminology and Examples ... I need help with yet another aspect of Example 1.14 ... ... Example 1.14 reads as follows: Near the end of the above text from T...
  40. Math Amateur

    MHB Example on Triangular Rings - Lam, Example 1.14

    I am reading T. Y. Lam's book, "A First Course in Noncommutative Rings" (Second Edition) and am currently focussed on Section 1:Basic Terminology and Examples ... I need help with an aspect of Example 1.14 ... ... Example 1.14 reads as follows: https://www.physicsforums.com/attachments/5984...
  41. Math Amateur

    I Example on Triangular Rings - Lam, Example 1.14

    I am reading T. Y. Lam's book, "A First Course in Noncommutative Rings" (Second Edition) and am currently focussed on Section 1:Basic Terminology and Examples ... I need help with an aspect of Example 1.14 ... ... Example 1.14 reads as follows: I cannot follow why the results in Table 1.16...
  42. Rectifier

    Efficient Method for Finding Units in Number Theory Rings (Z12, ⊗, ⊕)

    The problem Consider the ring ##(Z_{12}, \otimes, \oplus)## Find all units. The attempt I know that I am supposed to find units u such that ##gcd(12,u)=1## But how do I do it the easiest way? I am not very keen to draw a multiplication table, calculate the terms and search where the...
  43. Math Amateur

    MHB What are the ideals of the matrix ring?

    I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 4.2 Noetherian and Artinian Modules ... ... I need help with fully understanding Example 6 "Right Artinian but not Left Artinian" ... in Section 4.2 ... ... Example 6 reads as follows: My problem is...
  44. Math Amateur

    I Noncommutative Artinian Rings

    I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 4.2 Noetherian and Artinian Modules ... ... I need help with fully understanding Example 6 "Right Artinian but not Left Artinian" ... in Section 4.2 ... ... Example 6 reads as follows: My problem...
  45. Rectifier

    How Do You Find the Inverse of 5 in Z12?

    The problem Consider a ring ##(Z_{12}, \oplus, \otimes)## Find ##5^{-1}##. The attempt I am basically searching for an inverse to ##5^{-1}## (if I am correct then judging from previous examples in my book, 5 wouldn't qualify) Since rings are about residues (the set ##Z_{12}## includes residues...
  46. Math Amateur

    MHB Maximal Ideals and the Correspondence Theorem for Rings .... Rotman Proposition 5.9

    I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 5.1 Prime Ideals and Maximal Ideals ... I need some help with understanding the proof of Proposition 5.9 ... ...Proposition 5.9 reads as follows: In the proof of Proposition 5.9, Rotman...
  47. Math Amateur

    I Maximal Ideals and the Correspondence Theorem for Rings

    I am reading Joseph J. Rotman's book: Advanced Modern Algebra (AMA) and I am currently focused on Section 5.1 Prime Ideals and Maximal Ideals ... I need some help with understanding the proof of Proposition 5.9 ... ...Proposition 5.9 reads as follows: In the proof of Proposition 5.9, Rotman...
  48. 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.
  49. A

    Finding the Electric Field at the Midpoint of two rings

    Homework Statement Hello, Two 10-cm-diameter charged rings face each other, 20 cm apart. The left ring is charged to -22 nC and the right ring is charged to +22 nC . What is the magnitude of the electric field E⃗ at the midpoint between the two rings? Homework Equations [/B] E = Kq/r^2 K =...
  50. Q

    Why do waves form in a glass of water when hitting the table?

    So I have a cup of cold water placed on a table, as I hit the table with my palm I'm seeing concentric circles forming and disappearing on the water surface. Why is this happening? Are they waves or something else?
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