### Prove or disprove the existence of non-commutative division ring of order 4.

Prove or disprove the existence of non-commutative division ring of order 4.

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Prove or disprove the existence of non-commutative division ring of order 4.

Let F be the set of all functions f : R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

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

If f(x) = x^5 + 2x^3 + x^2 + 2x + 3, g(x) = x^4 + x^3 + 4x^2 + 3x + 3, then find greatest common divisor of f(x) and g(x) over the field of residue classes modulo 5 and express it in the form d(x) = m(x) f(x) + n(x) g(x) where d(x) = g.c.d. of f(x) and g(x).

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.

Given two elements a, b in the Euclidean ring R their least common multiple cЄR is an element in R such that a│c and b│c and such that whenever a│x and b│x for xЄR then c│x. Prove that any two elements in the Euclidean ring R have a least common multiple in R.

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

In a Euclidean ring R, any associate of a greatest common divisor is a greatest common divisor. keywords: GCD, GCDs

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 )

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

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

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

Let R be an integral domain, "m" a maximal ideal. Let S=R-m a) Prove that S^-1R is a local ring. b) Suppose R=Z, m=(5). Describe S^-1R.

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

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

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).

Modern Algebra Ring Theory (IX) The Field of Quotients of an Integral Domain Prove that the mapping φ:D→F defined by φ(a) =

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.

Modern Algebra Ring Theory (VII) The Field of Quotients of an Integral Domain

Prove that if [a , b] = [a΄, b΄] and [c , d] = [c΄, d΄] then [a , b][c , d] = [a΄, b΄][c΄, d΄].

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 Є R.

Can any set that is not a group (Z for example) still be a ring or is it necessary that a set must be a group to be a ring? Please give an example and counter example.

Let R be a ring with the property that every element is either nilpotent or invertible. If a, b, c are in R with a and b nilpotent, show that ac, ca, and a + b are nilpotent. For the latter, first observe that a + b cannot equal 1. Conclude that Nil (R) is the set of all nilpotent elements of R. (nil radical Nil (R) is define

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.

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 .

A1. Which of the following binary operations on R... A2. Solve each of the following equations for x in a group G with a, b, c.... A3. Define what is meant by an abelian group. Let GL2(R) be the group of non-singular 2 ×2 real matrices under multiplication. Decide whether or not each of the following subgroups of GL2(R) is ab

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

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

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

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