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    Ring Theory

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    Integral Domains with quotient fields

    ----------------- Let R1 and R2 be integral domains with quotient fields F1 and F2 respectively. If phi: R1 -> R2 is a ring isomorphism, show that phi extends to an isomorphism phi hat : F1 -> F2. Here extends means that phi hat(a) = phi (a) for all a in R1 (Hint: under the givien assumptions, there is really only one way to

    Abstract Algebra : Fields, Rings and Domains

    I) Show that if D is an integral domain then D[x] is never a field. ii) Is the assumption "D is an integral domain" needed here? That is, does the conclusion hold if D is merely assumed to be a ring?

    Abstract Algebra Proof : Ideals and Ring Homomorphisms

    Let and be ideals of the ring and suppose I C J. Prove: The function phi : R/I --> R/J defined by phi(a+I)=a+J is a well-defined ring homomorphism that is also onto. Please see the attached file for the fully formatted problem.

    Volume of Ring-Shaped Solid and Volume Expressed in Terms of Height

    A) A cylindrical drill with radius r1 is used to bore a hole through the center of a sphere of radius r2. Find the volume of the ring-shaped solid that remains. b) Express the volume in part (a) in terms of the height h of the ring. Notice that the volume depends only on h, not on r1 and r2.

    Rings and Subrings Functions

    Please see the attached file for the fully formatted problems. 1. Ler R be a ring, and , prove, using axioms for a ring, the following ? The identity element of R s unique ? That -r is the unique element of R such tht (-r)+r = 0. (hint, for part 1, suppose that 1 and 1' ate two identities of R, show that 1-1' must be

    Binomial Expansion in a Ring

    Let p be a prime. Show that in the ring Z-p (set of integers modulo p) we have (a+b)^p = a^p+b^p for all a, b in Z-p. The following hint was given: observe that the usual binomial expansion for (a+b)^n is valid in a commutative ring.

    Euler Tour : Dominoes

    2. A domino is a 2x1 rectangular piece of wood. On each half of the domino is a number, denoted by dots. In the figure, we show all C(5,2) = 10 dominoes we can make where the numbers on the dominoes are all pairs of values chosen from {1,2,3,4,5} (we do not include dominoes where the two numbers are the same). Notice that we hav

    Rings with Unity that Form a Group

    I need to prove the following: Show that if U is the collection of all units in a ring <R, +, *> with unity, then <U,*> is a group. A reminder was given to make sure to show that U is closed under multiplication.

    The Heat Equation on a Metal Ring

    Let u(x,t) describe the temperature of a thin metal ring with circumference 2pi. For convenience, let's orient the ring so that x spans the interval |-pi, pi|. Suppose that the ring has some internal heating that is angle-dependent, so that u(x, t) satisfies the inhomogeneous heat equation u_t = ku_zz + f(x), where k is t

    Ideals and Rings: Homomorphisms

    Problem: Prove the Second Isomorphism Theorem: If A is an ideal of R and S is a subring of R, then S+A is a subring, A, and (S intersecting A) are ideals of S+A and S, respectively, and (S+A)/A isomorphic to A/(S intersecting A).

    Ideals and Factor Rings : Locality

    Problem: A ring R is called a local ring if the set J(R) of nonunits in R forms an ideal. If p is a prime, show that Z(p) = {n/m belonging to Q | p does not divide m } is local. Describe J(Z(p))

    Ideals and Factor Rings: Prime Ideal

    Problem: Let R be a commutative ring. Show that every maximal ideal of R is prime and if R is finite, show that every prime ideal is maximal. ALSO is every prime ideal of Z(integers) maximal? Why?

    Ideals and Factor Rings: Annihilator

    Problem: If X is contained in R is a nonempty subset of a commutative ring R, define the annihilator of X by ann(X) = { a belonging to R | ax=0 for all x belonging to X} Show that X is contained in ann[ann(X)] AND Show that ann(X) = ann{ann[ann(X)]}.

    Ideal and factor rings

    Please if you are going to solve this problem please be very accurate!! (and careful!!). I have used brainmass before and have recieved incorrect answers on mulitple occations (answers that initially seemed correct) and it has hindered my studying. I have decided to give it another chance. With that said, thank you very much for

    Ideals and Factor Rings

    Problem: Note: | | is trying to denote a matrix If R = |S S| |0 S| and A = |0 S| |0 0| , S and ring, show that A is an ideal of R and describe the cosets in R/A

    Rings and Units

    Given r and s in a ring R, show that 1 + rs is a unit if and only if 1 + sr is a unit.

    Quotient Ring

    Please see the attached PDF file. I would prefer a solution in PDF format. Thanks!

    Presence of Ring Isomorphisms

    Please see the attached file for the fully formatted problem. Let gcd(m, n) = 1. We know that Zmn = Zm × Zn as an additive group. Is there a ring isomorphism Zmn = Zm × Zn? Same question for the complex numbers C with C = R × R as an additive group.

    Rings : Elements Obtained

    Please see the attached file for the fully formatted problems. Describe the ring obtained from Z12 by adjoining the element 1/2 (the inverse of 2).

    Rings : Fields

    Please see the attached file for the fully formatted problems. Prove that Z5/(x2 + x + 1) is a field. How many elements are there in this field? Can you also represent it as Z5[x]/(x2-a) where a is some element of Z5?

    Describe the Rings

    Please see the attached file for the fully formatted problem. Describe the rings: Z[x]/(x2 − 3, 2x + 4), Z[i]/(2 + i) where i2 = −1.

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