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Exercises
Confirm the statements made in Example 3 by proving that the following sets are subrings of the ring of all real numbers.
The set of all real numbers of the form
The set of all real numbers of the form
and
The set of all real numbers of the form
and
rational numbers.
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Chapter 5 Solutions
Elements Of Modern Algebra
- Confirm the statements made in Example 3 by proving that the following sets are subrings of the ring of all real numbers. The set of all real numbers of the form with and . The set of all real numbers of the form with and rational numbers. The set of all real numbers of the form , with and rational numbers.arrow_forwardExercises 2. Decide whether each of the following sets is a ring with respect to the usual operations of addition and multiplication. If it is not a ring, state at least one condition in Definition 5.1a that fails to hold. The set of all integers that are multiples of . The set of all real numbers of the form with and . The set of all real numbers of the form , where and are rational numbers. The set of all real numbers of the form , where and are rational numbers. The set of all positive real numbers. The set of all complex numbers of the form , where (This set is known as the Gaussian integers.) The set of all real numbers of the form with and . The set of all real numbers of the form with and .arrow_forwardTrue or False Label each of the following statements as either true or false where represents a commutative ring with unity. A polynomials in over is made up of sums of terms of the form where each and.arrow_forward
- 19. Find a specific example of two elements and in a ring such that and .arrow_forwardTrue or False Label each of the following statements as either true or false. The field of rational numbers is an extension of the integral domain of integers.arrow_forward50. Let and be nilpotent elements that satisfy the following conditions in a commutative ring: Prove that is nilpotent. for somearrow_forward
- True or False Label each of the following statements as either true or false. The characteristic of a ring is the positive integer such that for all in.arrow_forwardAn 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.arrow_forwardTrue or False Label each of the following statements as either true or false. 4. Both , the set of even integers, and, the set of odd integers, are subrings of the set of all integers.arrow_forward
- 37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.arrow_forward14. Let be the set of all real numbers of the form , where . Carry out the construction of the quotient field for this integral domain, and show that this quotient filed is isomorphic to the set of real numbers of the form where and are rational numbers.arrow_forwardLet a0 in the ring of integers . Find b such that ab but (a)=(b).arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,