3. Let (Mzz(2), +.) be the ring of all 2 x 2 matrices over a ring Z with the usualoperations of addition and multiplication on matrices.IrI- aE Z), then (1,+..) is:[a 01(a) Not left and not right ideal of (M(2), +.)(b) A left and net right ideal of (Ma(2), +.)(c) A right and not left ideal of (Mz2(2), +,)(d) A left and right ideal of (M2va(2), +.)

Answered: Image /qna-images/answer/95192db3-c20e-49c6-abaa-a019dc9172e2.jpg

3 Let (Mz(2), +.) be the ring of all 2x2 matrices over a ring Z with the usual operations of addition and multiplication on matrices. aeZ), then (1.+..) is: (a) Not left and not right ideal of (Mz(2), +..) (b) A left and net right ideal of (Ma(2), t) (c) Aright and not left ideal of (M2=z(2), +..) (4) A left and right ideal of (M2(Z), +.) lo ol

3 Let (Mz(2), +.) be the ring of all 2x2 matrices over a ring Z with the usual operations of addition and multiplication on matrices. aeZ), then (1.+..) is: (a) Not left and not right ideal of (Mz(2), +..) (b) A left and net right ideal of (Ma(2), t) (c) Aright and not left ideal of (M2=z(2), +..) (4) A left and right ideal of (M2(Z), +.) lo ol

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.1: Polynomials Over A Ring

Problem 18E: 18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is...

See similar textbooks

Similar questions

[Type here] Examples 5 and 6 of Section 5.1 showed that is a commutative ring with unity. In Exercises 4 and 5, let . 4. Is an integral domain? If not, find all zero divisors in . [Type here]
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]
a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].
An element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.
25. 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 to
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?
[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]
22. Let be a ring with finite number of elements. Show that the characteristic of divides .
18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .
Let R be a ring, and let x,y, and z be arbitrary elements of R. Complete the proof of Theorem 5.11 by proving the following statements. a. x(y)=(xy) b. (x)(y)=xy c. x(yz)=xyxz d. (xy)z=xzyz Theorem 5.11 Additive Inverses and Products For arbitrary x,y, and z in a ring R, the following equalities hold: (x)y=(xy) b. x(y)=(xy) (x)(y)=xy d. x(yz)=xyxz (xy)z=xzyz
19. Find a specific example of two elements and in a ring such that and .
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 .)