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# Ring Theory

### F[x]/(f(x)) is a field with p^n elements.

Modern Algebra Ring Theory (XXXII) Polynomial Ring Irreducible Polynomial If f(x) is in F[x], where F is the field

### F[x]/(x^2 + 1) is isomorphic to the field of complex numbers

Modern Algebra Ring Theory (XXXI) Polynomial Ring Irreducible Polynomial Let F be a field of real nu

### Polynomial Rings: Irreducible Polynomials

Prove that x^2 + 1 is irreducible over the field F of integers mod 11 and prove directly that F[x]/(x^2 + 1) is a field having 121 elements.

### Ring Theory : Abelian Groups, Nilradicals and Augmentation Ideals

Let p be a prime and let G be an abelian group of order a power of p. Prove that the nilradical of the group ring G is the augmentation ideal.

### Rings and Ideals

1.a) Let R be a ring with 1 and let S=M2(R). If I is an field of S, show that there is an ideal J of R such that I consists of all 2X2 matrices over J. 1 b) Use the result of 1 a) to prove the following question. Let R be the ring of 2X2 matrices over reals; suppose that I is an ideal of R. Show that I =(0) or I=R.

### Polynomial Rings : Relatively Prime Elements

Let F, K be two fields, F is a subset of K and suppose f(x), g(x) &#1028; F[x] are relatively prime in F[x]. Prove that they are relatively prime in K[x].

### Ring & Field Theory : Associative and Distributive Properties of Multiplication

Let F be the set of all functions f : R&#61664;R. We know that <F, +> is an abelian group under the usual function addition, (f + g)(x) = f(x) + g(x). We define multiplication on F by (fg)(x) = f(x)g(x). That is, fg is the function whose value at x is f(x)g(x). Show that the multiplication defined on the set F satisfies axio

### Ring Theory: Greatest Common Divisor (GCD)

Please respond to the following question: Find the greatest common divisor of the following polynomials over F, the field of rational numbers: x^3 - 6x^2 + x + 4 and x^5 - 6x + 1

### Ring Theory : Euclidean Rings and GCD

If the least common multiple of a and b in the Euclidean ring R is denoted by [a,b], prove that [a,b] = ab/(a, b) where (a, b) is the greatest common divisor of a and b.

### Euclidean Rings and Elements with Least Common Multiples

Given two elements a, b in the Euclidean ring R their least common multiple c&#1028;R is an element in R such that a&#9474;c and b&#9474;c and such that whenever a&#9474;x and b&#9474;x for x&#1028;R then c&#9474;x. Prove that any two elements in the Euclidean ring R have a least common multiple in R.

### Ring Theory : commutative ring with a unit element form abelian

Prove that the units in a commutative ring with a unit element form an abelian group.

### Ring Theory : Euclidean Rings

Prove that in a Euclidean ring ( a , b ) can be found as follows b = q0 a + r1 where d ( r1) < d (a ) a = q1 r1 + r2 where d ( r2) < d (r1 ) r1 = q1 r1 + r2 where d ( r3) < d ( r2 )

### Ring Theory : Direct Products, Mappings and Homomorphisms

Let I be a non-empty index set with a partial order <=, and A_i be a group for all i in I. Suppose that for every pair of indices i,j there is a map phi_ij:A_j ->A_i such that phi_jiphi_kj= phi_ki wheneveri<=j<=k, and phi_ii=1 for all i in I. Let P be the subset of elements(a_i) with i from I in the direct product D of A_i such

### Ring Theory: Direct and Inverse Limits, Homorphisms and Abelian Groups

Let I be a non-empty index set with a partial order<=. Assume that I is a directed set, that is, that for any pair i,j in I there is a,k in J such that i<=k and j<=k. Suppose that for every pair of indices i,j with i<=j ther is a map p_ij: A_i->A_j such that p_jkp_ij=p_ik whenever i<=j<=k and p_ii=1 for all i in I. Let B be the

### Surjective Ring Projections

Let m and n be positive integers with n dividing m. Prove that the natural surjective ring projection Z/mZ ->Z/nZ is also surjective on the units:(Z/mZ)^x ->(Z/nZ)^x

### Quotient Rings, Laurent Series and Power Series

If F is a field, prove that the field of fractions of F[[x]] is the ring F((x)) of formal Laurent series. Show that the field of fractions of the power series ring Z[[x]] is properly contained in the field of Laurent series Q((x)). Ps. Here F[[x]] is the ring of formal power series in the indeterminate x with coefficients in

### Homomorphism of Commutative Rings, Prime Ideals, Maximal Ideals

Let phi:R->S be a homomorphism of commutative rings a) Prove that if P is a prime ideal of S then either phi^-1(P)=R or phi^-1(P) is a prime ideal of R. Apply this to the special case when R is a subring of S and phi is the inclusion homomorphism to deduce that if P is a prime ideal of S then PR is either R or prime ideal in

### Ring Theory: Euclidean Ring

Prove that a necessary and sufficient condition that the element 'a' in the Euclidean ring is a unit is that d(a) = d(1). Or, Prove that the element 'a' in the Euclidean ring is a unit if and only if d(a) = d(1).

### Ring theory proof fields

Modern Algebra Ring Theory (IX) The Field of Quotients of an Integral Domain Prove that the mapping &#966;:D&#8594;F defined by &#966;(a) =

### Ring theory proof in modern algebra

Modern Algebra Ring Theory (VIII) The Field of Quotients of an Integral Domain Prove the distributive law in F , the field of quotients of D, where D is the ring of integers.

### Ring Theory : Division Rings

Let R be a ring such that the only right ideals of R are (0) and R. Prove that either R is a division ring or, that R is a ring with a prime number of elements in which ab = 0 for every a,b &#1028; R.

### Ring Theory/Largest two-sided ideal

Let I be a right ideal of a ring R and let A = {r in R: (R/I)r = 0}. Prove that A is the largest two-sided ideal of R contained in I.

### Units and Zero-divisor in a Commutative Ring with Identity

B8. Define what is meant by (a) a unit, (b) a zero-divisor in a commutative ring with identity. Show that an element in a communtative ring with identity (where 1 = 0) cannot be both a unit and a zero-divisor. Find an element in Z...Z which is neither a unit nor a zero-divisor. Show that... is a unit in .... Find a unit .

### Rings and Proofs for Commutative Rings

(See attached file for full problem description with proper symbols) --- 1A) Let R be a commutative ring and let A = {t &#61536; R &#61560; tp = 0R} where p is a fixed element of R. Prove that if k, m &#61536; A and b &#61536; R, then both k + m and kb are in A. 1B) Let R be a commutative ring and let b be a fixed ele

### 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

### 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.