Describe the kernel of epimorphism
Assume that each of
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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_forward12. Consider the mapping defined by . Decide whether is a homomorphism, and justify your decision.arrow_forward18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .arrow_forward
- 15. Let and be elements of a ring. Prove that the equation has a unique solution.arrow_forwardLet R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)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
- Exercises If and are two ideals of the ring , prove that is an ideal of .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_forwarda. 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 ].arrow_forward
- Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .arrow_forwardAssume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn Prove that is an epimorphism.arrow_forward22. Let be a ring with finite number of elements. Show that the characteristic of divides .arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,