Purchase Solution

Rings with Unity, Isomorphism, Bijectiveness and Invertibility

Not what you're looking for?

Ask Custom Question

Let R be a ring with unity e, R' a set, H a bijective map from R' into R. show that R' becomes a ring with unity if one defines:
a'+ b' = Hˉ¹ (H(a')+ H(b'))
a' b' = Hˉ¹(H(a') H(b'))
0'= Hˉ¹(0)
e'= Hˉ¹(e)
and that is an isomorphism of R' with R. 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.

Please see attached for full question.

Attachments
Purchase this Solution
Solution Summary

Rings with Unity, Isomorphism, Bijectiveness and Invertibility are investigated.

Solution Preview

Please see the attached file for the complete solution.
Thanks for using BrainMass.

Proof:
First, I show that is a ring. For any , we know and . Thus . So is the additive identity of . Since and , then we have . So is the multiplicative identity. It is clear that is ...

Purchase this Solution
Free BrainMass Quizzes
Solving quadratic inequalities

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

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.

Exponential Expressions

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

Probability Quiz

Some questions on probability

View More Free Quizzes
Related BrainMass Solutions

  • Rings, Commutative Rings, Idempotents, Subrings and Isomorphisms

    41110 Rings, Commutative Rings, Idempotents, Subrings and Isomorphisms (1) Given a ring R, an element e is called an idempotent if e^2 = e. (i) Let R1 and R2 be two commutative rings with unity. Consider R = R1 x R2.

  • Modules, Submodules, Commutative Rings with Unity and Homomorphism

    40194 Modules, Submodules, Commutative Rings with Unity 1. Let R and S be commutative rings with unity, and let φ: R S be a ring homomorphism.

  • Commutative rings

    This implies that there is a natural isomorphism between F(M⊗_A N) and F(M)⊗_BF(N). This solution helps with a question about communicative rings.

  • Ring theory proof

    Isomorphic Two rings are said to be isomorphic if there is an isomorphism of onto .

  • Structure of Rings

    Therefore, y7=y2 and y8=y3. Thus (y3+y2+y)(y5+1)=y8+y7+y6+y3+y2+y= y3+y2+y+ y3+y2+y= 2y3+2y2+2y This solution helps find the structure of rings.

  • subset

    239522 Isomorphism Subjective Subsets 1. Determine (up to isomorphism) all semisimple rings of order 1008. How many of them are commutative? (Recall that finite division rings are fields.) 2. A ring R is Boolean if x2 = x for all x  R.

  • Some concepts of groups and rings

    A mapping of rings is called a (ring) homomorphism if the following conditions are satisfied: (i) for any , we have and ; (ii) . Let : be an isomorphism of rings. Let be the following mappings: , for .

  • Abstract algebra: isomorphism

    So and thus is one-to-one. Third, I show that is onto. For any , we can find , such that . So is onto. Therefore, is a ring isomorphism. Hence . Done. This is a proof showing that given rings are isomorphisms.

  • Ideals and Rings : Homomorphisms

    17290 Ideals and Rings: Homomorphisms Problem: Prove the Second Isomorphism Theorem: If A is an ideal of R and S is a subring of R, then S+A is a subring, A, and (S intersecting A) are ideals of S+A and S, respectively, and (S+A)/A isomorphic to A/(

View More Related Solutions