Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 25322. Prove the Third Isomorphism Theorem for rings: Let R be a ring and I andJ be ideals of R, where J ⊂ I. ThenR/I ∼ = R/JI/J .23. Prove the Correspondence Theorem: Let I be a ideal of a ring R. Then S →S/I is a one-to-one correspondence between the set of subrings S containingI and the set of subrings of R/I. Furthermore, the ideals of R correspond toideals of R/I.24. Let R be a ring and S a subset of R. Show that S is a subring of R if andonly if each of the following conditions is satisfied.(a) S ≠ ∅.(b) rs ∈ S for all r, s ∈ S.(c) r − s ∈ S for all r, s ∈ S.25. Let R be a ring with a collection of subrings {R α }. Prove that ⋂ R α is asubring of R. Give an example to show that the union of two subrings cannotbe a subring.26. Let {I α } α∈A be a collection of ideals in a ring R. Prove that ⋂ α∈A I α is alsoan ideal in R. Give an example to show that if I 1 and I 2 are ideals in R,then I 1 ∪ I 2 may not be an ideal.27. Let R be an integral domain. Show that if the only ideals in R are {0} andR itself, R must be a field.28. Let R be a commutative ring. An element a in R is nilpotent if a n = 0 forsome positive integer n. Show that the set of all nilpotent elements forms anideal in R.29. A ring R is a Boolean ring if for every a ∈ R, a 2 = a. Show that everyBoolean ring is a commutative ring.30. Let R be a ring, where a 3 = a for all a ∈ R. Prove that R must be acommutative ring.31. Let R be a ring with identity 1 R and S a subring of R with identity 1 S .Prove or disprove that 1 R = 1 S .32. If we do not require the identity of a ring to be distinct from 0, we will nothave a very interesting mathematical structure. Let R be a ring such that1 = 0. Prove that R = {0}.33. Let S be a subset of a ring R. Prove that there is a subring R ′ of R thatcontains S.