Which of the following statement is false?(a) The characteristic of an integral domain iseither zero or a prime number(b) A skew field contains divisors of zero(c) The set of all non-zero elements of a skewfield forms a group with respect to multiplica-tion(d) A non-trivial finite ring without divisors ofzero is u skew field

Given are four statements in (a), (b), (c) and (d) To Choose the false statement

Answered: Which of the following statement is…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.3: The Characteristic Of A Ring

Problem 7E

See similar textbooks

Similar questions

Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.
Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros in
8. Prove that the characteristic of a field is either 0 or a prime.
Prove that if F is an ordered field with F+ as its set of positive elements, then F+nen+, where e denotes the multiplicative identity in F. (Hint: See Theorem 5.34 and its proof.) Theorem 5.34: Well-Ordered D+ If D is an ordered integral domain in which the set D+ of positive elements is well-ordered, then e is the least element of D+ and D+=nen+.
Consider the set ={[0],[2],[4],[6],[8]}10, with addition and multiplication as defined in 10. a. Is R an integral domain? If not, give a reason. b. Is R a field? If not, give a reason. [Type here][Type here]
Find all monic irreducible polynomials of degree 2 over Z3.
22. Let be a ring with finite number of elements. Show that the characteristic of divides .
[Type here] 15. Give an example of an infinite commutative ring with no zero divisors that is not an integral domain. [Type here]
Prove that if R and S are fields, then the direct sum RS is not a field. [Type here][Type here]
. a. Let, and . Show that and are only ideals of and hence is a maximal ideal. b. Show that is not a field. Hence Theorem is not true if the condition that is commutative is removed. Theorem 6.22 Quotient Rings That are Fields. Let be a commutative ring with unity, and let be an ideal of . Then is a field if and only if is a maximal ideal of .
Use Theorem to show that each of the following polynomials is irreducible over the field of rational numbers. Theorem Irreducibility of in Suppose is a polynomial of positive degree with integral coefficients and is a prime integer that does not divide. Let Where for If is irreducible in then is irreducible in .
If F is an ordered field, prove that F contains a subring that is isomorphic to . (Hint: See Theorem 5.35 and its proof.) Theorem 5.35: Isomorphic Images of If D is an ordered integral domain in which the set D+ of positive elements is well-ordered, then, D is isomorphic to the ring of all integers.

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS