Answered: Is there a ring homomorphism from the…

Math

Advanced Math

Is there a ring homomorphism from the reals to some ring whosekernel is the integers?

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.1: Ideals And Quotient Rings

Problem 35E: Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a...

See similar textbooks

Related questions

Q: Show that a unit of a ring divides every element of the ring.

A: Let us assume a unit in a ring R as a. Let us assume any element as x in that ring R. By using the…

Q: Theovem Za hes zero ducae ring PLme no The

A: Using definition of no zero divisor and prime number we prove given result.

Q: In each case, find the characteristic of the ring. (a) Zn x Zm (b) M₂(Zn) (c) Zx Zn

A: We know the definition of characteristics of ring, Let R be the ring then the characteristics of…

Q: Q1. a) Give an example of a semi-ring that is not a semi -algebra. b) Give an example of a…

Q: Let f (x) = x"+ 3x3+ 2x²+2 and %3D g(x) = * + 2x +1 E Zq [x]. Find g(x)g x(x) E Z5 [x] Such that…

Q: Give an example of a ring that has exactly two maximal ideals

A: To Do: Give an example of a ring that has exactly two maximal ideals.

Q: are subrings of R, show that SOT is a subring. Is this the case for +tse S, te T}?

Q: 6. If a and b are not zero divisors in a ring R, prove that ab is not a zero divisor.

Q: R= 7 Z. Show that the polynomial (4,0)x in R[x] has infinitely meny roots in R.

A: Given : R = ℤ ⊕ ℤ To show that the polynomial (1, 0)x in R[x] has infinitely many roots in R.

Q: A commutative ring R is said to be Artinian if there is no infinite decreasing chain of ideals in R,…

A: Since, you have posted a multiple question. But according to guidelines we will solve the first…

Q: Show that the set of all rational numbers is only a subring but not an ideal of the ring of real…

Q: In a ring R without unity, show that every idempotent is a zero divisor but is not nilpotent.

Q: Show that the nilpotent elements of a commutative ring form a subring

A: We have to prove that the nilpotent elements of a commutative ring form a subring. In order to do…

Q: Describe all the subrings of rings of the integers

A: We have to describe the rings and subrings of integers.

Q: The number of zero divisors in the ring Z80

A: Step:-1 Theorem:- If gcda, n>1 and n- a then [a] is a zero divisor in ℤn. Also, we know ϕn =…

Q: Please use method from Abstract Algebra, Show that the polynomial 37a² + 30a³ + a⁴ = 1500 does not…

Q: Explain why the ring of integers Z under usual addition and multiplication is not a field.

A: Given that the set of integers Z is a ring under addition and multiplication. We know that an…

Q: Suppose that a and b belong to a commutative ring and ab is a zero-divisor.Show that either a or b…

Q: Check if the following subsets of real numbers are Rings or not. • {a + b*(2)^(1/3) | a, b ∈ Q} • {a…

Q: The number of zero divisors of the ring Z4 O Z, is

Q: Give an example of a noncommutative ring with unity.

Q: 2/32 a) = Z5 52/32 쓴 Z X R Z × R, 4Z × {0} b)

A: We have to show that the following are isomorphic: and . and . First Isomorphism Theorem: Let…

Q: Q3: Prove that the ring of rational numbers (Q, +,.) is division ring

Q: Is the multiplicative inverse of an element in a polynomial ring unique? Or can there be more than…

A: This can be proved as shown below:

Q: What is a ring homomorphism in simple terms?

Q: Exercise 3. Consider the structure (Z+, *) of the positive integers with multipli- cation. Show that…

A: We will first prove that any set A is definable on the structure (Z+,*) iff : Either P⊆ A or, A∩ P…

Q: Show that the set Q + Q √ 3 is a field. Q denotes set of all rational numbers

Q: I is set f all integers (positic, is the ordinaiy multiplication negative and zero),. and show that…

Q: Suppose that R is a commutative ring without zero divisors . show that characterstic of R is 0 or…

A: suppose that R is a commutative ring without zero divisors to show- characteristic of Ring R is 0 or…

Q: Find all the primitive elements of Galois field GE (16) , where the monic irreducible poly nomial is…

A: In field theory, a primitive element of a finite field GF(q) is a generator of the multiplicative…

Q: Is the collection of irrational numbers R\ Qa field? Why or why not?

Q: 5. Give an example where a and b are not zero divisors in a ring R, but the sum a +b is a zero…

Q: let R- Cla.b] (condhreveer Faubioe peterucel set of al ato b HaR) że R is real greaeber Volued Cort…

Question

Is there a ring homomorphism from the reals to some ring whose
kernel is the integers?

Expert Solution

Step 1

To Do:

We have to check whether there is a ring homomorphism from the reals to some ring whose kernel is the integers.

Step by step

Solved in 2 steps

Knowledge Booster

$Ring$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

[Type here] 15. Give an example of an infinite commutative ring with no zero divisors that is not an integral domain. [Type here]
37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.
Prove that a finite ring R with unity and no zero divisors is a division ring.
7. Prove or disprove that the set of nonzero integers is closed with respect to division defined on the set of nonzero real numbers.
32. Consider the set . a. Construct addition and multiplication tables for, using the operations as defined in . b. Observe that is a commutative ring with unity, and compare this unity with the unity in . c. Is a subring of ? If not, give a reason. d. Does have zero divisors? e. Which elements of have multiplicative inverses?
Consider the set S={ [ 0 ],[ 2 ],[ 4 ],[ 6 ],[ 8 ],[ 10 ],[ 12 ],[ 14 ],[ 16 ] }18. Using addition and multiplication as defined in 18, consider the following questions. Is S a ring? If not, give a reason. Is S a commutative ring with unity? If a unity exists, compare the unity in S with the unity in 18. Is S a subring of 18? If not, give a reason. Does S have zero divisors? Which elements of S have multiplicative inverses?
10. Prove or disprove that the set of all nonzero integers is closed with respect to a. addition defined on . b. multiplication defined on .
7. Prove that on a given set of rings, the relation of being isomorphic has the reflexive, symmetric, and transitive properties.
11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .
22. Let be a ring with finite number of elements. Show that the characteristic of divides .
An element a of a ring R is called nilpotent if an=0 for some positive integer n. Prove that the set of all nilpotent elements in a commutative ring R forms a subring of R.
14. Letbe a commutative ring with unity in which the cancellation law for multiplication holds. That is, if are elements of , then and always imply. Prove that is an integral domain.