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Ring Theory

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).

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.