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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

What is an algebraic ring?

Could you please break down what is an algebraic ring? I need this explained in laymans terms so that I will understand.

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

Rings and fields

(a) If R is a field, show that R itself is a field of fractions for R. (b) Show that Q is a field of fractions for Z and for 2Z. please see attached pdf.

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