Purchase Solution

Rings and Subrings

Not what you're looking for?

Ask Custom Question

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
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

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Solving quadratic inequalities

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

Probability Quiz

Some questions on probability