# Rings and Subrings

Not what you're looking for?

Please see the attached file for the fully formatted problems.

1. 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.

(hint, for part 1, suppose that 1 and 1' ate two identities of R, show that 1-1' must be zero, and for part 2, suppose that there is an element such that s+r = 0, and prove that s = -r)

2. let R be the set of complex 4th roots of 1. so R = {1,-1,i,-i} . Does R, together with the usual addition and multiplication of complex numbers, form a ring? Justify your answer.

3.

? Let R be the ring . Show that is a subring but not an ideal of R.

? Let R be a ring. Define what is meant by a polynomial over R in the inderminate x.

4. let and be polynomials in . Calculate f + g and fg in .

##### Purchase this Solution

##### Solution Summary

Rings and Subrings are investigated. The solution is detailed and well presented.

##### Solution Preview

Please see the attached file for the full solution.

Thanks for using BrainMass.

1. Proof:

(a) Suppose and are two identities of . Then for any , we have and . Then . Especially, when , we have . This implies . Therefore, the identity of is unique.

(b) Suppose there is another element , such that ...

##### Purchase this Solution

##### Free BrainMass Quizzes

##### Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

##### Exponential Expressions

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

##### 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.

##### Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

##### Probability Quiz

Some questions on probability