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Ring Theory

Ring Homomorphism

See attached pdf file. --- - Find all ring homomorphisms from Z... ---

Maximal Ideals, Residues and Ring Homomorphisms

Let . Show that the map the residue of a+ b modulo 2, is a ring homomorphism with . Prove that . Hence, or otherwise, give a maximal ideal of . Consider the ideal (2)+(x) of . Show that (2)+(x) . Hence explain why (x) is not a maximal ideal of . NOTE: All question marks are Z, the integers Please see the attached

Ring Homomorphisms, Residues and Distinct Maximal Ideals

Let . Prove that the map given by , where is the residue of a modulo n, is a ring homomorphism. Find the kernel and image of . Prove that if is a ring homomorphism, then given by is also a ring homomorphism. Write down two distinct maximal ideals of . Does have a finite or infinite number of maximal ideals?

Rings and fields

(See attached file for full problem description with proper symbols) For part one....the first is in rational numbers, and second is in integers. --- ? Verify that is a sub field of and that is a sub ring of . ? Let R be a commutative ring and I an ideal of R. let and be elements of R. prove that is then

Rings, Commutative Rings, Idempotents, Subrings and Isomorphisms

(1) Given a ring R, an element e is called an idempotent if e^2 = e. (i) Let R1 and R2 be two commutative rings with unity. Consider R = R1 x R2. Find two non-zero idempotents e1,e2 E R such that 1= e1 + e2 and e1e2 = 0. (Be careful: what is 1 in this R? What is 0?) (ii) On the other hand, suppose R is any commutative ring wit

Ideal rings

Note: Z is integer numbers C is set containment Here is the problem Let I be an ideal in a ring R. Define [ R : I ] = { r in R such that xr in R for all x in R } 1) Show that [ R : I ] is an ideal of R that contains I 2) If R is assumed to have a unity, what can you say about [ R : I ] ? 3) Find [ 2Z

Modules, Submodules, Commutative Rings with Unity and Homomorphism

1. Let R and S be commutative rings with unity, and let φ: R S be a ring homomorphism. If M is an S-module, prove that M is also an R-module if we define rm = φ(r)m for all r E R and m E M. 2. If M1 and M2 are submodules of an R-module M such that M = M1(+)M2, prove that M1 = M/M2 and M2 = M/M1.

Rings and Modules : Quasi-Regular, Module Homomorphisms and Kernels

1. Let R be a ring. Prove that if x, y E R such that xy is right quasi-regular then yx is also right quasi-regular. 3. Let M and N be left R-modules. Let f : M N and g : N M be left R-module homomorphisms such that fg(y) = y for all y N. Show that M = ker(f) im(g). Please see attached.

Ring Identities

For any x,y E R (i.e. Ring R) the following equalities hold. a) 0.x=0 b) a(-b)=(-a)b=-(ab) Prove either a or b. State any properties used in your proof. Please see attached for full question.

Ideals of a Ring : Containment and Subgroups

Let J and I be ideals of the ring R, with J C I C R. Show that J is an ideal in the ring I (Recall that any ideal of a ring is also a subring; so I is a ring in its own right) Note that: To show J an ideal of I, we must show that 1) it is an additive subgroup of R and 2) it is satisfied bJ C J and Jb C J

Rings of Unity, Monoid, Momomorphism and Invertible Elements

Let R commutative ring with unity, and S a sub monoid of the multiplicative monoid of R. In RxS define (a,b) ~ (b,t) if Эu є S э u(at-bs)= 0. Show that ~ is an equivalence relation in RxS. Denote the equivalence class of (a,s) as a/s and the quotient set consisting of these classes as RS-¹. Show that RS-¹ be

Rings with Unity, Isomorphism, Bijectiveness and Invertibility

Let R be a ring with unity e, R' a set, η a bijective map from R' into R. show that R' becomes a ring with unity if one defines: a'+ b' = ηˉ¹ (η(a')+ η(b')) a' b' = ηˉ¹(η(a') η(b')) 0'= ηˉ¹(0) e'= ηˉ¹(e) and that is an isomorphism of R' with R. Use

Ring Homomorphisms

Let phi be a homomorphism of a ring R with unity onto a nonzero ring R'. Let u be a unit in R. Show that phi (u) is a unit in R'.

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.

Rings and Subrings

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

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

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 : 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 : 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 contiained in ann[ann(X)] AND Show that ann(X) = ann{ann[ann(X)]}

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!

Rings : Elements

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).