### Ring Isomorphisms

Prove that Z[X] and Q[X] are not isomorphic.

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Prove that Z[X] and Q[X] are not isomorphic.

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Find a commutative ring R with no ideal other than (0) and R that is not a Field.

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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.

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

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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 def

(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 e

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

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