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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)


(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.

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

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

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

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

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

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

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

Ring and ideal proofs

** Please see the attached file for the complete problem description ** Let R be a ring....Prove that radical I is an ideal.....Prove that there is a prime ideal P containing I....


1. a. show that every subfield of complex numbers contains rational numbers b. show that the prime field of real numbers is rational numbers c. show that the prime field of complex numbers is rational numbers 2. a. Let R be a domain. Prove that the polynomial f(x) is a unit in R[x] if and only if f(x) is a nonzero cons

Logic and validity of arguments

Hi Shrikant, I wonder if your can help me with this: 1) Determine whether the argument is valid or invalid. A tree as green leaves or the tree does not produce oxygen. This tree has green leaves. ------------------------------------------------------------- Therefore:. This tree does not produce oxyge

Sqrarefree Integers, Fields, Conductors and Maximal Ideals

Let D be a squarefree integer, and let 0 be the ring of integers in the quadratic field Q(sqrtD). For any positive integer f prove that the set Of = Z[fw] = {a + bfw | a, b E Z} is a subring of 0 containing the identity. Prove that [O:Of]= f (index as additive abelian groups). Prove conversely that a subring of 0 containing the

Commutative ring with no non-zero nilpotent elements

Let R be a commutative ring with no non-zero nilpotent elements (that is, a^n = 0 implies a = 0). If f(x) = a_0 + a_1x + a_2x^2 +...+ a_mx^m in R[x] is a zero-divisor, prove that there is an element b is not equal to 0 in R such that ba_0 = ba_1 = ba_2 = ...=ba_m = 0. See attached file for full problem description.

Show that a set of matrices is a ring without an identity element.

I am given the set of infinite-by-infinite matrices with real entries that have only finitely many nonzero entries. Also, the matrix has entries a_ij, where i and j are natural numbers and there is a natural number n such that a_ij = 0 if i >= n or j >= n. I need to understand how to show that this set of matrices is a ring w