Below are two sets of real numbers. Exactly one of these sets is a ring, with the usual addition and multiplication operations for realnumbers. Select the one which is a ring.O[a+bn:a, bez}O[a+b√2: a, b e Z}Let R be the ring above. True or falseR is a ring with identity.OTrueOFalseR is a skewfield.OTrueOFalseR is a commutative ring.OTrueOFalse

Using the definitions of a ring we solve the above problem. The detailed solution typeset in LaTEX…

Answered: Below are two sets of real numbers.…

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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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?
24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)
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.
22. Define a new operation of addition in by and a new multiplication in by. a. Is a commutative ring with respect to these operations? b. Find the unity, if one exists.
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}.
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 .
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.
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.
If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.
Given that the set S={[xy0z]|x,y,z} is a ring with respect to matrix addition and multiplication, show that I={[ab00]|a,b} is an ideal of S.
An element x in a ring is called idempotent if x2=x. Find two different idempotent elements in M2().
19. Find a specific example of two elements and in a ring such that and .

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