Rings Definition and 437 Threads

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...

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...

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...

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...

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...

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...

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)...

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...

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...

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...

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.

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...

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?

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...

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...

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.

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...

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?

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...

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...

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...

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

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...

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...

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...

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?

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...

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...

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...

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...

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...

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...

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...

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...

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) ...

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.

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...

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...

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)...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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, +...

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