Purchase Solution

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

Not what you're looking for?

Ask Custom Question

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 without identity element.

Purchase this Solution
Solution Summary

It is shown that a set of matrices is a ring without an identity element. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

Solution Preview

Proof:
Suppose the set of infinite-by-infinite matrices with real entries that have only finite many nonzero entries is R.
First, I show that R is a ring.
For any two elements A=(a_ij) and B=(b_ij), we have the following observations.
1. A+B=(a_ij+b_ij)=(b_ij+a_ij)=B+A is also an infinite-by-infinite matrix. Since A and B have finite many nonzero entries, assume A has m nonzero entries and B has n nonzero entries, then A+B has at ...

Purchase this Solution
Free BrainMass Quizzes
Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

Probability Quiz

Some questions on probability

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

View More Free Quizzes
Related BrainMass Solutions

  • Rings and groups

    Let [equationB] be the ring of n x n matrices with entries in R. What is the identity element of S?

  • Sets and Rings

    Multiplicative identity: There exists an element such that for all , 1*a=a*1=a (a ring satisfying this property is termed a unit ring, or sometimes a "ring with identity"), 9.

  • Rings

    (3) To show that a subset of a ring is a *subring*, you need only show that: i.

  • Elements in an Abelian group

    Think of elements of End_Z(G) as infinite matrices with integer entries. Definition: Let G be an abelian group and let End_Z(G) be the set of all group homomorphisms from G to itself. We call it the endomorphism ring of G.

  • set of all invertible diagonal matrices

    393561 set of all invertible diagonal matrices Let R be any ring with identity 1 and GL(n, r) the group of invertible nxn matrices over R and SL(n, R) those matrices of determinant 1. 1) Show that SL(n, R) forms a normal subgroup of GL(n, R) 2) Define

  • Rings and Subrings

    Ler R be a ring, and , prove, using axioms for a ring, the following ? The identity element of R s unique ? That -r is the unique element of R such tht (-r)+r = 0.

  • Algerbraic Structures: Units and Zero-divisor in a Commutative Ring with Identity

    Show that an element in a communtative ring with identity (where 1 = 0) cannot be both a unit and a zero-divisor. Find an element in Z...Z which is neither a unit nor a zero-divisor. Show that... is a unit in .... Find a unit ...

  • Rings with Unity, Isomorphism, Bijectiveness and Invertibility

    Use this to prove that if u is an invertible element of a ring then ( R, +, .u ,0,uˉ¹), where a .u b =aub is a ring is ring isomorphic to R. Show also that (R, θ ,o, 1, 0), where a θ b = a+b -1, a o b= a+b -ab is ring isomorphic to R.

  • Rings and Ideals

    Thanks for using BrainMass. 1.a) Let R be a ring with 1 and let S= . If I is an ideal of S .show that there is an ideal J of R such that I consists of all 2X2 matrices over J. Let x= be an element of I.

View More Related Solutions