Question

Define a+b=a+b -1 and a*b=ab-(a+b)+2 Assume that (Z, +,*) is a ring. (a) Prove that the...

Define a+b=a+b -1 and a*b=ab-(a+b)+2 Assume that (Z, +,*) is a ring. (a) Prove that the additative identity is 1? (b) what is the multipicative identity? (Make sure you proe that your claim is true). (c) Prove that the ring is commutative. (d) Prove that the ring is an integral domain. (Abstrat Algebra)

Math   Advanced-Math   
0 0
Add a commentTranscribed image text
Next >

Homework Answers

Answer #1

0 0
Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Log In

Sign Up

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
If for integers a, b we define a ∗ b = ab + 1, then: (a)...
If for integers a, b we define a ∗ b = ab + 1, then: (a) The operation ∗ is commutative ? (b) The operation ∗ is associative ? Modern Algebra
If for integers a, b we define a ∗ b = ab + 1, then: (a)...
If for integers a, b we define a ∗ b = ab + 1, then: (a) The operation ∗ is commutative ? (b) The operation ∗ is associative ? Modern Abstract Algebra, please explain, thanx
Let f : Z → Z be a ring isomorphism. Prove that f must be the...
Let f : Z → Z be a ring isomorphism. Prove that f must be the identity map. Must this still hold true if we only assume f : Z → Z is a group isomorphism? Prove your answer.
1. Consider the set (Z,+,x) of integers with the usual addition (+) and multiplication (x) operations....
1. Consider the set (Z,+,x) of integers with the usual addition (+) and multiplication (x) operations. Which of the following are true of this set with those operations? Select all that are true. Note that the extra "Axioms of Ring" of Definition 5.6 apply to specific types of Rings, shown in Definition 5.7. - Z is a ring - Z is a commutative ring - Z is a domain - Z is an integral domain - Z is a field...
Prove or disprove (a) Z[x]/(x^2 + 1), (b) Z[x]/(x^2 - 1) is an Integral domain. By...
Prove or disprove (a) Z[x]/(x^2 + 1), (b) Z[x]/(x^2 - 1) is an Integral domain. By showing (a) x^2+1 is a prime ideal or showing x^2 + 1 is not prime ideal. By showing (b) x^2-1 is a prime ideal or showing x^2 - 1 is not prime ideal. (Hint: R/I is an integral domain if and only if I is a prime ideal.)
Assume a, b ∈ Z and p is prime. Using B´ezout’s identity, prove that if p...
Assume a, b ∈ Z and p is prime. Using B´ezout’s identity, prove that if p | ab, then p | a or p | b.
Prove that (A+B)^2 = A^2 + 2AB + B^2 is true if and only if (AB)^-1...
Prove that (A+B)^2 = A^2 + 2AB + B^2 is true if and only if (AB)^-1 = A^-1B^-1.
Let R = Z with addition ⊕ and multiplication ⊗ defined as follows: a ⊕ b...
Let R = Z with addition ⊕ and multiplication ⊗ defined as follows: a ⊕ b := a + b − 1 a ⊗ b := ab − (a + b) + 2 Show that this a commutative ring with unity
4. Let a, b, c be integers. (a) Prove if gcd(ab, c) = 1, then gcd(a,...
4. Let a, b, c be integers. (a) Prove if gcd(ab, c) = 1, then gcd(a, c) = 1 and gcd(b, c) = 1. (Hint: use the GCD characterization theorem.) (b) Prove if gcd(a, c) = 1 and gcd(b, c) = 1, then gcd(ab, c) = 1. (Hint: you can use the GCD characterization theorem again but you may need to multiply equations.) (c) You have now proved that “gcd(a, c) = 1 and gcd(b, c) = 1 if and...
Let R be a ring, and assume that r^2 = r for all r ∈ R....
Let R be a ring, and assume that r^2 = r for all r ∈ R. (a) Prove that r + r = 0 for all r ∈ R. (b) Prove that R is commutative.
ADVERTISEMENT
Need Online Homework Help?

Get Answers For Free
Most questions answered within 1 hours.

Ask a Question
ADVERTISEMENT