Answer Happy

DER ER 1 Determine if each of the following statements is True or False. (a) Every subgroup of Z7 x 214 is abelian. (b) There exists a group H such that the product group H x Sz is cyclic. (c) The ring Z23[2] is a principal ideal domain. (d) Every field is an integral domain. (e) Let R1, R, be rings. Then any ideal of the direct product Rị x R2 is of the form 11 x 12 := {(Q1, 42)|a1 € lj, az € 12} for some ideals I CR1,1, C R2. (f) If R is an integral domain, then for any ideal I of R, the quotient ring R/I is necessarily an integral domain. (g) Let a, b be integers. If the ring Z/aZ is isomorphic to Z/bZ, then (a) = (b). (h) Every one-to-one ring homomorphism from Z3[2] to itself is surjective. (i) Let A, B be finite groups. Let o: A + B be a group homomorphism. If ker o consists only of the identity element of A, then the order of B must be greater than or equal to the order of A. () For all f, 9 € Q, there exists h € Z[x] such that the ideal (1,9) generated by f,9 in Q[2) is equal to the principal ideal (h) generated by h in Q[r].