Below are two sets of real numbers. Exactly one of these sets is a ring, with the usual addition and multiplication operations for real numbers.Select the one which is a ring.O{a/2:a € Z}O{a/2:a e Z, neN}Let R be the ring above. True or false:R is a ring with identity.OTrueOFalseR is a skewfield.OTrueOFalseR is a commutative ring.OTrueOFalse

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.1: Definition Of A Ring

Problem 54E

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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?
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?
a. 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}.
21. Define a new operation of addition in by with a new multiplication in by. a. Verify that forms a ring with respect to these operations. b. Is a commutative ring with respect to these operations? c. Find the unity, if one exists.
An element x in a ring is called idempotent if x2=x. Find two different idempotent elements in M2().
37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.
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 .
19. Find a specific example of two elements and in a ring such that and .
Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4
22. Let be a ring with finite number of elements. Show that the characteristic of divides .
If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.
41. Decide whether each of the following sets is a subring of the ring. If a set is not a subring, give a reason why it is not. If it is a subring, determine if is commutative and find the unity, if one exists. For those that have a unity, which elements in have multiplicative inverses in? a. b. c. d. e. f. g. h.

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