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Ring Theory

Commutative Rings, Subrings and Submodules

Problem: I need to show that (i) leads to (ii), then (ii) leads to (iii): Let S be a commutative ring, R be a subring in S and x be an element from S. Show that the following are equivalent: (i) There exist from R where such that ; In other words x is a root of normalized polynomial over R. (ii) Submodule R - modu

Complex Numbers, Integer and Rational Coefficient

Consider the ring Z[i] of complex numbers with integer coefficients. Prove or disprove that the field of fractions associated with Z[i] is isomorphic to Q[i], the field of complex numbers with rational coefficients. keywords: isomorphisms

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.

Ring Homomorphisms and Ideals

Let : R->Q be a ring homomorphism , and suppose that I is a non-trivial ideal of R. Prove or disprove that (I)={ (i)| i I } is necessary an ideal of Q. Let : R->Q be a ONTO ring homomorphism , and suppose that I is a non-trivial ideal of R. Prove or disprove that (I)={ (i)| i I } is necessary an ideal of Q. Plea

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)

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: Determining the 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 )