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Ring Theory

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 : 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

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 : 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.

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

Ring Homomorphism

See attached pdf file. --- - Find all ring homomorphisms from Z... ---

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 :