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_forwardExercises 10. Prove Theorem 5.4:A subset of the ring is a subring of if and only if these conditions are satisfied: is nonempty. and imply that and are in .arrow_forwardExercises Prove Theorem 5.3:A subset S of the ring R is a subring of R if and only if these conditions are satisfied: S is nonempty. xS and yS imply that x+y and xy are in S. xS implies xS.arrow_forward
- Let I be the set of all elements of a ring R that have finite additive order. Prove that I is an ideal of R.arrow_forwardExercises Let I be a subset of ring R. Prove that I is an ideal of R if and only if I is nonempty and xy, xr, and rx are in I for all x and yI, rR.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_forward
- 21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.arrow_forward18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .arrow_forwardAn element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.arrow_forward
- 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=xzyzarrow_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_forwardLet R be a ring. Prove that the set S={ xRxa=axforallaR } is a subring of R. This subring is called the center of R.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,