Assume that the set
Verify that the mapping
Describe ker
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Elements Of Modern Algebra
- 14. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.arrow_forward15. Let and be elements of a ring. Prove that the equation has a unique solution.arrow_forwardAssume that the set R={[x0y0]|x,y} is a ring with respect to matrix addition and multiplication. Verify that the mapping :R defined by ([x0y0])=x is an epimorphism from R to Z. Describe ker and exhibit an isomorphism from R/ker toarrow_forward
- 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 .)arrow_forwardFind the characteristic of each of the following ring: a. b. c. M2() d. M2() e. M2(2) f. M2(3)arrow_forward14. Let be an ideal in a ring with unity . Prove that if then .arrow_forward
- Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).arrow_forwardComplete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .arrow_forwardLet :312 be defined by ([x]3)=4[x]12 using the same notational convention as in Exercise 9. Prove that is a ring homomorphism. Is (e)=e where e is the unity in 3 and e is the unity in 12?arrow_forward
- 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.arrow_forward18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .arrow_forwardAssume R is a ring with unity e. Prove Theorem 5.8: If aR has a multiplicative inverse, the multiplicative inverse of a is unique.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Elementary Linear Algebra (MindTap Course List)AlgebraISBN:9781305658004Author:Ron LarsonPublisher:Cengage Learning