Consider the set
are real
numbers, with the same rules for addition and multiplication as in
a. Show that
b. Show that
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Elements Of Modern Algebra
- Let R be the set of all matrices of the form [abba], where a and b are real numbers. Assume that R is a commutative ring with unity with respect to matrix addition and multiplication. Answer the following questions and give a reason for any negative answers. Is 12 an integral domain? Is R a field?arrow_forwardGiven 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.arrow_forwardLet S be the set of all 2X2 matrices of the form [x0x0], where x is a real number.Assume that S is a ring with respect to matrix addition and multiplication. Answer the following questions, and give a reason for any negative answers. Is S a commutative ring? Does S have a unity? If so, identify the unity. Is S an integral domain? Is S a field? [Type here][Type here]arrow_forward
- 15. Let and be elements of a ring. Prove that the equation has a unique solution.arrow_forward28. a. Show that the set is a ring with respect to matrix addition and multiplication. b. Is commutative? c. does have a unity? d. Decide whether or not the set is an ideal of and justify your answer.arrow_forwardConsider the matrices below. X=[1201],Y=[1032],Z=[3412],W=[3241] Find scalars a,b, and c such that W=aX+bY+cZ. Show that there do not exist scalars a and b such that Z=aX+bY. Show that if aX+bY+cZ=0, then a=b=c=0.arrow_forward
- 19. Find a specific example of two elements and in a ring such that and .arrow_forwardLet 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+y4arrow_forwardAn element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.arrow_forward
- Examples 5 and 6 of Section 5.1 showed that P(U) is a commutative ring with unity. In Exercises 4 and 5, let U={a,b}. Is P(U) a field? If not, find all nonzero elements that do not have multiplicative inverses. [Type here][Type here]arrow_forward25. Figure 6.3 gives addition and multiplication tables for the ring in Exercise 34 of section 5.1. Use these tables, together with addition and multiplication tables for to find an isomorphism from toarrow_forward[Type here] Examples 5 and 6 of Section 5.1 showed that is a commutative ring with unity. In Exercises 4 and 5, let 6. Let where and are the elements of. Equality, addition, and multiplication are defined in as follows: if and only if and in , a. Prove that multiplication inis associative. Assume thatis a ring and consider these questions, giving a reason for any negative answers. b. Isa commutative ring? c. Doeshave a unity? d. Isan integral domain? e. Isa field? [Type here]arrow_forward
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