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Ring Theory

Ring theory originated from Fermat's last theorem and is a subtopic of abstract algebra that studies rings. Rings are defined to be an abelian group that includes both operations of addition and multiplication. A set R is a ring if it satisfies the associative property, distributive property and has a multiplicative identity. The associative property for multiplication would be satisfied if:

(a x b) x c = a x (b x c)

holds true for all a, b, and c where a, b, and c are elements of the ring. The distributive property would be satisfied if the following equality were satisfied similarly:

a x (b + c) = (a x b) + (a x c)

and

(b + c) x a = (b x a) + (b x c)

Furthermore, there must exist an element 'y' such that for all elements of the ring such as 'a' this is true:

y x a = a x y = a

This is what is meant by having a multiplicative identity. If only these three properties were satisfied then the ring would be a non-commutative ring. In order for the ring to be a commutative ring it would also have to satisfy the commutative property. That is to say

a x b = b x a

holds true for all the elements of the ring. The above formulations are for the operation of multiplication. Similar properties for addition would also have to be satisfied. A common example of some rings are the set of integers, the set of rational numbers, the set of real numbers, and the set of complex numbers.

R-modules

Suppose R is a ring, G, M are R-modules and Hom(G,M) is the set of R-module homomorphisms from G to M. Identify Hom(Z/nZ,Z), Hom(Z,Z/nZ), Hom(Z/3Z,Z/6Z), Hom(Z/10Z,Z/6Z) as abelian groups, where n belongs to Z and Z is the set of integers.

Simple Ring: Artenian Ring

Please help with the following problem. A simple ring is an Artenian ring with no 2-sided ideals. Let K be a field. Let V be an n-dimensional vector space over K and let A = EndK(V), the ring of n x n matrices over K. The ring is clearly Artenian since it has finite dimension over K. We want to see that the 2-sided ideal are

Proving Local Rings

i) Show that a non-zero ring R is local (i.e. it has a unique maximal ideal) if and only if x or 1 - x is a unit, where x is a member of R. ii) Show why or why not are the following four rings local: {a/b belongs to Q s.t. b is not divisible by a prime p}, Z[x]/(X^3), F[[x]] (where F[[x]] is the ring formed by the set of all

Product Rings and Relations

• In each of the cases below, describe the ring obtained from F2 by adjoining an element x satisfying the relation: (i) x2+x+1=0, (ii) x2+1=0, (iii) x2+x=0. • Determine if Z/(6) is isomorphic to Z/(2)xZ/(3).

Ideals and Maximal Ideals

Please help with the following problems. a) If I,J are ideals in a ring R such that I+J=R and R is isomorphic to the product ring (R/I)x(R/J) when IJ=0, describe the idempotents corresponding to this product decomposition; b) Describe the maximal ideal of RxR where in this case R is the set of real numbers; c) How many roo

Polynomial Rings and prime integers

Let p be any prime integer. Consider polynomials f(x) and g(x) of the form: f(x) = x^p g(x) = x over the finite field Zp. Prove that f(c) = g(c) for all c in Zp. Hint: Consider the multiplicative group of nonzero elements of Zp.

Ring Theory: Polynomial Rings

Consider the polynomial ring R=Q[x]. (a) show that I = {f(x) (x^3-6x+7)+g(x) (x+4) | f(x), g(x) in R} is an ideal of R. (b) We have seen that R is a principle ideal domain. That is, every ideal is generated by a single element of R. Find h(x) in R so that I = {f(x)h(x) | f(x) in R}.

The Problems in Ring Theory

When f1, ....fn are elements in the polynomial ring R[x, y], we denote by (f1,....,fn) the set { ∑gifi | g ∈ R[x, y]} which is an ideal of R[x, y] Let R be a commutative ring and let I ⊆ R be an ideal. Show that √I := { f ∈ R | there exists n ∈ N such that fn ∈ I } is an ideal of R. Let V(I) be defined as

Rationals with an odd denominator form a ring

Consider Q, the set of all rational numbers. For qâ??Q, q=m/n for m, n â?? Z. Suppose this is written in reduced form so gcd(m,n)=1. Let R â?? Q consist of all rational numbers for which the denominator of the reduced form is odd. Show that this set forms a ring under the usual operations of addition and multiplication.

Ring theory

1. If E, F are fields and F is a subring of E, show each q in Aut(E/F) permutes the roots in E of each nonzero p(x) in F[x]. Hint if p(x)=a0+a1x+a2x^2+. . . +anx^n then p(x) has at most n roots in E. show that for z in E, p(z)=0 implies p(q(z))=0 2 If R is a commutative ring of prime characteristic p, show the function f:R

Set of Ring Isomorphisms

Suppose F, E are fields and F is a subring of E. Prove that the set of ring isomorphisms Q:E-->E is a group Aut(E) under composition *, and that the set Aut(E/F) of isomorphisms Q in G, with Q(f)=f for all f in F is a subgroup of Aut(E). If E is a field and H is a subgroup of Aut(E), show that the set E^H of elements of E t

Evaluation Homomorphism - Ring Theory

Suppose F, E are fields and F is a subring of E. Suppose g is an element of E and is algebraic over F, that is p(g)=0 for some nonzero polynomial p(x) in F[x]. Then there must be a nonzero polynomial m(x) smallest degree among those nonzero polynomials in F[x] with g as a root. Prove m(x) cannot be factored as a(x)b(x) for poly

Ring Theory and Cartesian Product

If R and S are rings, the cartesian product RxS is a ring too with operations (r1,s1) + (r2,s2) = (r1+r2,s1+s2) (r1,s1)*(r2,s2) = (r1*r2,s1*s2) identity elements 0RxS = (0R,0s) 1RxS = (1R,1S) and additive inverse -(r,s) = (-r,-s) If R and S are nontrivial rings, show that RxS has at least 4 idempotent elemen

Prove that a finite subring of a field is a field.

Prove that a finite subring R of a field F is itself a field. Hint: if x is an element of R and x is not equal to 0 show the function f:R->R with f(r) = xr is injective. From finiteness of R, deduce that its image includes 1

Ring Theory: Maximization Help

9. The Acme Class Ring Company designs and sells two types of rings: the VIP and the SST. They can produce up to 24 rings each day using up to 60 total man-hours of labor. It takes 3 man-hours to make one VIP ring, versus 2 man-hours to make one SST ring. How many of each type of ring should be made daily to maximize the com

Groups, Rings and Fields: Example Questions

Indicate which of the following statements about sets under the specified operations are true. For the ones that are false, provide one counter-example. (a) The set of irrational numbers under addition forms an abelian group. (b) The set of complex numbers under multiplication forms a group. (c) The set of rational numbers

Advanced Algebra: Commutative Ring

Let x and y belong to a commutative ring R with prime characteristic p. a) Show that (x + y)^p = x^p + y^p b) Show that, for all positive integers n, (x + y)^p^n = x^p^n + y^p^n. c) Find elements x and y in a ring of characteristic 4 such that (x + y)^4 != x^4 + y^4.

Binomial Probability: Meeting the quota of 3 diamond rings

Dan must sell 3 diamond rings this week to meet his quota. He is meeting up with 5 possible customers, each who wants a different ring. If he has a 30% chance of making the sale with each customer, what is the probability that he will meet his quota by tomorrow. Now suppose 3 of his customers want the same ring, and they wil

Finding Target Area: Example Questions

An archer is to fire arrows at a target of concentric circles. The target - maker wishes to calculate the cross - sectional areas of the rings A, B, C and D of the target to make their areas in the ration of 4:3:2:1. The whole target is of radius 10 cm. (i) Calculate the fraction of the total area of the target filled by each

Commutative Ring Proofs

See the attached file. Let R be any commutative ring and S a subset of R n f0g containing no zero divisors. Let X be the Cartesian product R  S and denote a relation  on X where (a; b)  (c; d). (a) Show that  is an equivalence relation on X. (b) Denote the equivalence class of (a; b) by a=b and the set of equivalence cla

Finite Ring Proofs

Please see the attached pdf. I need a detailed, rigorous proof of this with explanation of the steps so I can learn. Let R be a finite ring. a. Prove that there are positive integers m and n with m>n such that x^m for ever x E R. b. Give a direct proof (i.e. without appealing to part c) that if R is an integral domain, the

Local ring proof

A local ring is commutative with identity which has a unique maximal ideal. Prove that R is local if and only if the non-units of R form an ideal. I need a detailed, rigorous proof of this with explanation of the steps so I can learn. Thank you.

Group and Ring Homomorphisms

Please attached for the question to help me understand Group and Ring Homomorphisms. Is this function a Homomorphism? a) Does 1:22 :22 where 1(a1,a2) = (a1,0) define a group homomorphism? a ring homomorphism? Prove your answers. b) Does 1:22 :22