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# Ring Theory

### Finitely Generated Z-modules

Definition: Let R be a commutative ring with identity, let M be an R-module, and let B be a nonempty subset of M. Then the set RB is defined as RB is a submodule. If B is a finite set, say , we write for RB, and say that RB is a finitely generated R-module. In particular, if for some , we say that M is finitely generate

### Ring Theory : Prove that an R-module M is Noetherian

Prove that an R-module M is Noetherian if and only if every submodule of M is finitely generated.

### Ring isomorphism

(See attached file for full problem description) Z2 means modulo 2 and Z2n means modulo 2n.

### Ring homomorphism

(See attached file for full problem description) --- 1. Show that if is a ring homomorphism and A is an ideal of R Then need not be an ideal of S. (Compare with property "If A is an ideal and is onto S, then is an ideal).

### Rings, Prime Ideals and Maximal Ideals

1.Give a example of a commutative ring that has a maximal ideal that is not a prime ideal. 2. Prove that I=<2+2i> is not a prime ideal of Z[i]. How many elements are in Z[i]/I ? What is the characteristic of Z[i]/I ? Please see the attached file for the fully formatted problems.

### Polynomial Rings : Prove Zero - Divisor

Let A be a commutative ring with identity 1, and let A[x] be the ring of polynomials with coefficients in A. Let f (x) = a0+ a1x + ... + anxn! A[x] . Prove that f (x) is a zero - divisor in A[x] IF AND ONLY IF (iff) 7c ! A, c ! 0 " cf (x) = 0. Please see the attached file for the fully formatted problems.

### Commutative Rings, Ideals, Kernels, Matrices and Injective and Surjective Ring Homomorphisms

If n &#1028; R and R is a commutative ring we indicate by Mn(R) the ring of allnxn entries wrt the usual operations on matrices. If n>1 this ring is commutative even if R is. Let S={(aij)&#1028;Mn(R)|i&#8800;j=>aij=0} Let k be an integer 1&#8804;k&#8804;n. Show that a) S is a commutative subring of Mn(R) b) The function f: S

### Maximal Ideals : Commutative Rings and Pointwise Operations ( Function Addition and Composition )

Let R be the set of all continuous functions from the set of real numbers into itself. Then R is a commutative ring with the following operations: (f+g)(x)=f(x) + g(x) and (fg)(x)=f(x)g(x) for all x. Now let I be the set of all functions f(x) an element of R such that f(1)=0. Show that I is a maximal ideal of R. Maximal

### Ideals: Prime Ideals and Commutative Ring

Show that N is contained in P for each prime ideal, P of a commutative ring R. Where N is the set of all nilpotent elements. "a" is nilpotent if a^n=0 for some positive integer n. N itself is an ideal.

### Ring Homomorphism

See attached pdf file.

### 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 ?

### Give an Example of a Ring which is Not a Principle Ideal Domain

Give an example of a ring which is not a principle ideal domain.

### Polynomial Rings, Finite Dimensional Vector Space

Prove that a module over a polynomial ring C[t] is a finite dimensional vector space with a linear operator that plays the role of multiplication by t.

### Ideals: Cyclic Module in a Commutative Ring

Prove that every cyclic module in a commutative ring R is of the form R/L for some ideal L. //Note: R/L is said "R modulo L."

### 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

1. Let R and S be commutative rings with unity, and let &#966;: R S be a ring homomorphism. If M is an S-module, prove that M is also an R-module if we define rm = &#966;(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 : Von Neumann Regularity and Jacobson Radicals

An element a of a ring R with unity is von Neumann regular if there exists b such that aba = a. Show that the only element in Jacobson radical J[R] having this property is the zero element.

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

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 Equality Properties

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

### Division subring S = division ring D or S is a subset of D

Modern Algebra Division Ring

### 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 &#1069;u &#1108; S &#1101; 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, H a bijective map from R' into R. show that R' becomes a ring with unity if one defines: a'+ b' = Hˉ¹ (H(a')+ H(b')) a' b' = Hˉ¹(H(a') H(b')) 0'= Hˉ¹(0) e'= Hˉ¹(e) and that is an isomorphism of R' with R. Use this to prove that if u is an invertible element of a ring

### Use Polar Coordinates to Find the Volume of a Ring-Shaped Solid

A cylindrical drill with radius 2 is used to bore a hole throught the center of a sphere of radius 7. Find the volume of the ring shaped solid that remains using polar coordinates.

### 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'.

### 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