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. S

    Correspondence theorem for rings

    Hello, "What does the correspondence theorem tell us about ideals in Z[x] that contain x^{2} + 1? My thinking is that since Z[x]/(x^{2} + 1) is surjective map and its kernel is principle and generated by x^{2} + 1 since x^{2} + 1 is irreducible. This implies ideals that contain x^{2} + 1 are...
  2. J

    Polynomial Rings: Finding 8 Elements with r^2=r

    Homework Statement Find eight elements r \in \mathbb{Q}[x]/(x^4-16) such that r^2=r. Homework Equations N/A The Attempt at a Solution The elements 0+(x^4-16) and 1+(x^4-16) clearly satisfy the desired properties, but I still need six more elements. Can anyone help me figure out a...
  3. A

    Rotating rod with little rings that slides out

    Homework Statement A uniform rod of mass M and length L rotates in horizontal plane about a fixed axis through its center and perpendicular to the rod. Two small rings, each with mass m are mounted so that they can slide along the rod. Rod rotates with the initial angular speed of ω...
  4. F

    How Does Water Alter the Diameter of Newton's Rings?

    Homework Statement Newton's rings can be seen when a planoconvex lens is placed on a flat glass surface. For a particular lens with an index of refraction of n= 1.51 and a glass plate with an index of n= 1.78, the diameter of the third bright ring is 0.760 mm. If water (n= 1.33) now fills...
  5. L

    Factoring polynomial over rings

    Homework Statement Factor x^3+3x+6 over Z5, Z10, Z, Q and ℝ. 2. The attempt at a solution For Z5, I have the roots of x^3+3x+1 in Z5, x=2=-3 and x=3=-4, so x+3 and x+4 are factors. By long division of x^3+3x+1, it is found that x+3 is a repeated factor, so x^3+3x+6=(x+4)(x+3)^2. For Z and...
  6. ArcanaNoir

    Proving the Distributive Property for Rings

    Homework Statement I need to show that (-x)*y=-(x*y) for a ring. unless it's not true. Homework Equations A ring is a set R and operations +, * such that (R, +) is an Abelian group, * is associative, and a*(b+c)=a*b+a*c and (b+c)*a=b*a+c*a. The Attempt at a Solution I don't know...
  7. G

    Show that a f: Z -> R , n -> n*1(subr) is a homomorphism of rings

    Homework Statement Show that a f: Z → R , n → n*1R is a homomorphism of rings Homework Equations The Attempt at a Solution I'm not sure how to exactly go about answering this question, but I'm going to try to start with the definition: f(a+b) = f(a) + f(b) f(a*b) = f(a) * f(b)...
  8. W

    Principal ideals of rings without unity

    Both my book and lecturer have in the definition a ring omitted the requirement of a unity. I was reading in my book about ideals, more specifically principal ideals. I stumbled over a formula that differed by whether or not the ring had a unity. As an example I state the two for principal left...
  9. D

    Induced magnetic field in rings

    Hello, Say I have two concentric conducting rings, where r1 >> r2 (why is this important, btw?), and I run a time alternating current I(1) thru the larger one. This will create a magnetic field B (also) thru the smaller ring, which in turn will create itself a magnetic field B2 and so on...
  10. Z

    Primes and Associates in Rings

    Homework Statement Let a, b be members of a commutative ring with identity R. If a is a a prime and a, b are associates then b is also prime. True/False Homework Equations Definitions: a is prime if a|xy implies a|x or a|y a and b are associates if there exists a unit u s.t a=bu...
  11. W

    News Time to sell those wedding rings.

    http://www.bloomberg.com/news/2011-08-22/gold-tops-1-900-for-first-time-on-concern-global-economy-may-weaken-more.html" Prices are still said to rise. Up and down up and down.
  12. K

    Rings as Algebras: Definitions & Questions

    I'm trying to make sense of two different definitions of an algebra over a ring. The definitions are as follows: If R is a commutative ring, then 1) S is an R-algebra if S is an R-module and has a compatible ring structure (such that addition agrees) 2) If \alpha:R \to S is a ring...
  13. H

    Semilocal rings vs Picard group

    For a dimension 1 regular semi-local ring, does the Picard group vanish? What if it is not regular? (and what if I ask for the ideal class group?) What if it's dimension greater than 1?
  14. C

    Why the plane of the rings is vertical to the rotation axis of Saturn?

    Why the plane of the rings is vertical to the rotation axis of Saturn? I don't think this is a coincidence, because the orbits of the 8 planets in our Solar System is roughly in a plane, and some galaxies are like a plate, so does the accretion plane of the black hole. But as we know...
  15. F

    Rings of Electric Field and Gauss's Law

    Homework Statement Trying to find the E-field inside a conductor using rings even though my book tells me it is 0. I haven't learned how to do surface integrals yet but I think I only need Calc II to do this. The Attempt at a Solution...
  16. D

    Rings Isomorphism: Proving R & R_2 Subrings of Z & M_2(Z)

    3. Let R = a+b \sqrt{2} , a,b is integer and let R_{2} consist of all 2 x 2 matrices of the form [\begin{array}{cc} a & 2b \\ b & a \\ \end{array} }] Show that R is a subring of Z(integer) and R_{2} is a subring of M_{2} (Z). Also. Prove that the mapping from R to R_{2} is a isomorphism.
  17. A

    Exploring Smoke Rings: Investigating Project Questions

    Hi all, I'm trying to answer the following questions related to a project on smoke rings. We are in an introductory class, so this project is more for fun and just applying the little we have learned about point, rigid body, and fluid motion to something less understood like vortex rings...
  18. R

    First isomorphism theorem for rings

    Consider: \varphi:R\rightarrow S is a homomorphism. Also,\hat{\varphi}:\frac{R}{ker\varphi}\rightarrow \varphi(R). How can I show \hat{\varphi} is bijective? Most textbooks say it is obvious. I see surjectivity obvious but not injectivity. Could anyone provide a proof for injectivity?
  19. FtlIsAwesome

    Exploring Rings of Worlds: Exo-Ring Systems & Moon Rings

    I've been thinking about rings. More massive objects would more likely be able to have rings. I think we'll eventually find superEarths, and even subEarths, that have rings. All the gas planets in our solar system have rings, but only Saturn has prominent ones. Exo-ring systems could vary...
  20. C

    Are zero divisors in R[x] also zero divisors in R?

    Let R be a commutative ring. If an doesn't equal 0 and a0+a1x+a2x^2+...+anx^n is a zero divisor in R[x], prove that an is a zero divisor in R. What I did was say if the polynomial is a zero divisor in R[x] then let that polynomial equal p(x) and any other polynomial be q(x) with...
  21. C

    Is an Non-Zero Coefficient in a Polynomial a Zero Divisor in R[x]?

    Let R be a commutative ring. If an doesn't equal 0 and a0+a1x+a2x^2+...+anx^n is a zero divisor in R[x], prove that an is a zero divisor in R. What I did was say if the polynomial is a zero divisor in R[x] then let that polynomial equal p(x) and any other polynomial be q(x) with...
  22. K

    Is the Direct Sum of Two Nonzero Rings Ever an Integral Domain?

    Show that the direct sum of 2 nonzero rings is never an integral domain I started by thinking about what a direct sum is (a,b)(c,d)=(ac,bd) (a,b)+(c,d)=(a+c,b+d) We have an integral domain if ab=0 implies a=0 or b=0
  23. I

    Newton rings: spherical droplet with thin film interference

    Homework Statement So I have a similar situation with the traditional Newton rings. Instead of having however a convex lens, I have a (partial) spherical surface with height d with sitting on top of a thin film of thickness t, with d >>> t, and refractive index n and below the thin film is a...
  24. M

    Ring homomorphisms of polynomial rings

    Homework Statement Let R be a commutative ring and let fa: R[x] -> R be evaluation at a \in R. If S: R[x] -> R is any ring homomorphism such that S(r) = r for all r\in R, show that S = fa for some a \in R. Homework Equations The Attempt at a Solution I don't get this at all...
  25. D

    Circular Interference, Newton's Rings

    Homework Statement The figure below shows a lens with radius of curvature R lying on a flat glass plate and illuminated from above by light with wavelength λ. This photo, taken from above the lens, shows that circular interference fringes (called "Newton's rings") appear, associated with the...
  26. A

    Why Carbon Atoms Don't Form Square or Triangular Rings

    I have heard of carbon compounds in the form of rings or circles like cyclohexane and cyclopentane and it set me wondering of why carbon atoms do not form square or triangular rings. Could someone explain the reason?
  27. M

    Onto Homomorphisms of Rings

    Homework Statement f: R -> R1 is an onto ring homomorphism. Show that f[Z(R)] ⊆ Z(R1) Homework Equations The Attempt at a Solution I'm a little confused. So f is onto, then for all r' belonging to R1, we have f(r)=r' for some r in R. But if f is onto couldn't R1 has less...
  28. C

    Abstract algebra questions relating to Ideals and cardinality of factor rings

    Homework Statement Find the number of elements in the ring Z_5[x]/I, where I is a) the ideal generated by x^4+4, and b) where I is the ideal generated by x^4+4 and x^2+3x+1. Homework Equations Can't think of any. The Attempt at a Solution I started by finding the zeros of the...
  29. W

    Quality of Lenses Using Newton's rings

    Homework Statement Basically I'm writing up a formal report on our experiment on Newton's rings - one of the things in our aims was in investigate the quality of the lenses we were using. Whilst many sites also say Newton's rings can be used to investigate the quality of the lenses not much...
  30. M

    Rings- units, nilpotents, idempotents

    Homework Statement Find the units, nilpotents and idempotents for the ring R = [\Re \Re] [0 \Re] (Those fancy R's are suppose to be the set of Reals by the way.. not good with this typing math stuff) Homework Equations The Attempt at a Solution I'm not actually sure I...
  31. J

    What causes the stability and formation of smoke rings and air rings in water?

    Take a look at these videos: Smoke rings in air: http://www.swisseduc.ch/stromboli/etna/etna00/etna0005video2-it.html?id=14 Air rings in water: http://www.youtube.com/watch?v=bT-fctr32pE&feature=related Second one is really amazing, as it shows two things: - great stability of the ring, which...
  32. D

    Benzene forms the electron shell configuration of two rings of

    Benzene forms the electron shell configuration of two rings of electrons, parallel to the molecule, shown here http://upload.wikimedia.org/wikipedia/commons/thumb/9/90/Benzene_Orbitals.svg/750px-Benzene_Orbitals.svg.png so my question is, if an electron beam is run through the center of the...
  33. N

    Polynomial Rings/Fields/Division Rings

    Homework Statement Let F be a field, F[x] the ring of polynomials in one variable over F. For a \in F[x], let (a) be all the multiples of a in F[x] (note (a) is an ideal). If b \in F[x], let c(b) be the coset of b mod (a) (that is, the set of all b + qa, where q \in F[x]). F[x]/(a), then is the...
  34. S

    Basic Newton's Rings Numerical Problem

    In Newton's Rings Experiment, what will be the order of the dark ring which will have double the diameter of that of the 20th dark ring? Wavelength(lambda = 5890 Angstrom); Radius of curvature is not given. Thin film is made of air, so refractive index is 1. And the light is incident normally...
  35. S

    Showing two rings are not isomorphic

    Homework Statement Explain why Z4 x Z4 is not isomorphic to Z16. Homework Equations Going to talk about units in a ring. Units are properties preserved by isomorphism. The Attempt at a Solution We see the only units in Z4 are 1 and 3. So the units of Z4 x Z4 are (1,1) , (3,3) ...
  36. J

    Simple Rings: Commutativity and Identity

    Hello everyone, i was checking out a paper on simple rings http://www.imsc.res.in/~knr/RT09/sssrings.pdf and they said that all commutative simple rings are fields. i just don't see why they should have identity. thank you.
  37. A

    Paradromic Rings: What Are My Creations Classified As?

    Attached are 3 of the images I created from a simple basic Moebius Band in the ChaosPro 3.3 In each one; I severed the band at one point and re attached it at another. Max x was positive Min x, Max y, and Min y were negative My X and Y axis rotations were also both negative. The images...
  38. T

    Calculate Radius of Curvature for Newton's Rings - 5th & 15th Bright Ring

    Homework Statement The diameter of the 5th and 15th bright ring formed by sodium light (wavelength = 589.3nm) are 2.303mm and 4.134mm respectively. Calculate the radius of curvature. Homework Equations DON'T ASSUME THAT THE RINGS ARE PERFECTLY CENTRED (I assume that means that the...
  39. silvermane

    Proving a Product of Commutative Rings not an Integral Domain:

    Homework Statement Prove that if we have two commutative rings R and S and form the product R X S, then R X S cannot be an integral domain. The Attempt at a Solution We have that an integral domain is a commutative ring with 1 not= 0 and with non-zero zero-divisors. ==> (1,0)X(0,1)...
  40. J

    Abstract Algebra, rings, zero divisors, and cartesian product

    The problem states: Let R and S be nonzero rings. Show that R x S contains zero divisors. I had to look up what a nonzero ring was. This means the ring contains at least one nonzero element. R x S is the Cartesian Product so if we have two rings R and S If r1 r2 belong to R and s1 s1...
  41. G

    Left ideals in simple rings

    I read the following on the wikipedia page about simple rings (http://en.wikipedia.org/wiki/Simple_ring): I do not see why this is the case. Take the ring of 3 by 3 matrices over the real numbers and the left ideal, J, generated by: \begin{pmatrix} 0 &1 &1\\ 0 &0 &0\\ 0 &0 &0...
  42. C

    Rings and Fields - Write down the nine elements of F9

    Rings and Fields - Write down the nine elements of F9[i] Homework Statement In F9 = Z/3Z, there is no solution of the equation x^2 = −1, just as in R. So “invent” a solution, call it 'i'. Then 'i' is a new “number” which satisfies i^2 = −1. Consider the set F9[i] consisting of all...
  43. P

    Carbocation Stability on Fused Rings

    So I just had a question on a quiz (did not go well) about carbocation stability on the fused rings bicyclo[2.2.1]heptane, with the positive charge on a bridgehead carbon and 2-methylbicyclo[2.2.1]heptane with the positive charge on C2. The question was which is more stable and why? The...
  44. D

    Explore Dependence of Axioms for Rings & Commutative Rings

    Homework Statement This is not an assignment question, just something that I am wondering about as an offshoot of an assignment question. In my course notes Rings are defined as having 3 axioms and commutative rings have 4.(outined below) I have just answered this question: Show that the...
  45. cronxeh

    Men Who Wear Rings: Finger Placement & Preferences

    So I've had this nice ring for a while, its the 'This too shall pass' kind of ring given to king Solomon by one of his wisemen. I also have a titanium band ring for my pinky, and a sterling silver blue sapphire skull ring. What fingers do you prefer? I have it on my left pinky, left middle...
  46. B

    Understanding Determinants of Matrices over Commutative Rings

    There is a small outline in a book about finding the determinant of a matrix over an arbitrary commutative ring. There are a few things I don't understand; here it is: 'Let R be a commutative ring with a subring K which is a field. We consider the matrix X = (x_{ij}) whose entries are...
  47. O

    Rings and Fields: Understanding Polynomials as a Commutative Ring with Unity

    Homework Statement is the set of all polynomials a ring,and a fieldd.Is is commutative and does it have unity Homework Equations The Attempt at a Solution now if we add or multiply any polynomials we get a polynomial. So it is a ring, but i am not sure what the multiplicative...
  48. S

    Reducing Ketones on Benzene Rings with H2, Pd/C, 45 psi

    Is there anyway that a Ketone on a benzene ring could be reduced to a hydrocarbon with the use of H2, Pd/C, 45 psi ? I know that carbonyls on aromatic rings are able to be reduced to alcohols via this method but would be interested in if it is possible (more so then practical) to fully reduce to...
  49. I

    Understanding Rings: Defining Addition and Multiplication in Abstract Algebra

    I need to learn some abstract algebra, and it's pretty hard doing this on my own. Please help me. According to the definition, Ring is an algebraic structure with two binary operations , commonly called addition (+) and multiplication ( . ). We write (R,+, .). Some examples of rings are: (Z, +...
  50. G

    Rings, fields, spaces etc.

    They all seem to be defined as sets with multiplication and addition axioms satisfied. What is the difference?
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