Exercises
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
and
The set of all real numbers of the form
and
are rational numbers.
The set of all real numbers of the form
and
are rational numbers.
The set of all positive real numbers.
The set of all
(This set is known as the Gaussian integers.)
The set of all real numbers of the form
and
The set of all real numbers of the form
and
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Chapter 5 Solutions
Elements Of Modern Algebra
- 37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.arrow_forward14. 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.arrow_forwardConsider 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?arrow_forward
- 12. Let be a commutative ring with prime characteristic . Prove, for any in that for every positive integer .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_forward50. Let and be nilpotent elements that satisfy the following conditions in a commutative ring: Prove that is nilpotent. for somearrow_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_forwarda. For a fixed element a of a commutative ring R, prove that the set I={ar|rR} is an ideal of R. (Hint: Compare this with Example 4, and note that the element a itself may not be in this set I.) b. Give an example of a commutative ring R and an element aR such that a(a)={ar|rR}.arrow_forwardTrue or false Label each of the following statements as either true or false. A ring homomorphism from a ring To a ring must preserve both ring operations.arrow_forward
- An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.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_forwardLet R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,