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Ring Theory

Maximal Ideals, Cosets, Polynomials and Quotient Rings

Consider the ring of Z[x] and its ideal (2, ) Find the size of Z[x] / (2, ) and find a coset representative for each coset of Z[x] / (2, ) ; Is the Z[x] / (2, ) a field (You need to prove it) ? Is the Z[x] / (2, ) an integral domain (You need to prove it)

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 )

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)

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 : 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 and Maximal Ideals

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.

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 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 : Matrix Proof

Prove that if [a , b] = [a&#900;, b&#900;] and [c , d] = [c&#900;, d&#900;] then [a , b][c , d] = [a&#900;, b&#900;][c&#900;, d&#900;].

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.

Sets and Rings

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.

Ring Theory/Nil radical

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 def

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 e

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

Zero Divisors

We just learned of homomorphisms, and zero divisors. How does knowing if an integer is one-to-one allow us to prove it to be a zero divisor?

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).