Rings of Unity, Monoid, Momomorphism and Invertible Elements
Not what you're looking for?
Let R commutative ring with unity, and S a sub monoid of the multiplicative monoid of R. In RxS define (a,b) ~ (b,t) if Эu є S э u(at-bs)= 0. Show that ~ is an equivalence relation in RxS. Denote the equivalence class of (a,s) as a/s and the quotient set consisting of these classes as RS-¹. Show that RS-¹ becomes a ring with unity relative to
a/b +b/t = (at +bs)/st
(a/s)(b/t) = (ab)(st)
Additive unity = 0/e
Multiplicative unity = e/e
Show that a --> a/e is homomorphism of R into RSˉ¹ and this is a monomorphism if and only if no element of S is a zero divisor in R.
Show that the elements s/e, s Є S, are units (set of invertible elements) in RS-¹.
Purchase this Solution
Solution Summary
Rings of Unity, Monoid, Momomorphism and Invertible Elements are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.
Solution Preview
Please see the attached file for the complete solution.
Thanks for using BrainMass.
Proof:
First, I show that ~ is an equivalent relation in . From the definition, we know that if and only if we can find some , such that . We need to verify the following three conditions.
1) Reflexive: Because is commutative, then . So for any , we have . Thus .
2) Symmetric: . Since , we can find some , such that . This implies ...
Purchase this Solution
Free BrainMass Quizzes
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Probability Quiz
Some questions on probability
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
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.