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# Ring Theory

### 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?

### {n.1 | n is interger} is a subdomain of the integral domain

Show that if D is an integral domain, then {n.1 | n is interger} is a subdomain of D contained in every subdomain of D.

### 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 Factor Rings: Annihilators and Fermat's Theorem

If R is a commutative ring, a polynomial f(x) in R[x] is said to annihilate R if f(a) = 0 for every a belonging to R Show that x^p - x annihilates Zp (Z is integers).

### 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.