Concept explainers
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
The set of all real numbers of the form
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Chapter 5 Solutions
Elements Of Modern Algebra
- 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.arrow_forwardLet a0 in the ring of integers . Find b such that ab but (a)=(b).arrow_forwardTrue or false Label each of the following statements as either true or false. 7. For the quotient ring of by the ideal is .arrow_forward
- 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?arrow_forwardLabel each of the following statements as either true or false. The field of rational numbers is complete.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
- Exercises 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_forward36. Suppose that is a commutative ring with unity and that is an ideal of . Prove that the set of all such that for some positive integer is an ideal of .arrow_forward21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,