Question

2. (a) Show that every ideal in ring Z is principal. More specifi-cally, prove the following: if A is an ideal in Z, then A = (n) = nZ, where n is the smallest positive integer in A, or A = {0}, if A contains no positive integers. [Remark: Integral domains in which every ideal is principal are called principal ideal domains. In addition to Z, an example of a principal ideal domain is R[x].] (b) Suppose that F is a field and there is a ring homomorphism from Z onto F. Show that F is isomorphic to Zp, for some prime p.

          2. (a) Show that every ideal in ring Z is principal. More specifi-cally, prove the following: if A is an ideal in Z, then
A = (n) = nZ,
where n is the smallest positive integer in A, or A = {0}, if A contains no positive integers. [Remark: Integral domains in which every ideal is principal are called principal ideal domains. In addition to Z, an example of a principal ideal domain is R[x].]

(b) Suppose that F is a field and there is a ring homomorphism from Z onto F. Show that F is isomorphic to Zp, for some prime p.

2. (a) Show that every ideal in ring Z is principal. More specifi-cally, prove the following: if A is an ideal in Z, then
A = (n) = nZ,
where n is the smallest positive integer in A, or A = 0, if A contains no positive integers. [Remark: Integral domains in which every ideal is principal are called principal ideal domains. In addition to Z, an example of a principal ideal domain is R[x].]

(b) Suppose that F is a field and there is a ring homomorphism from Z onto F. Show that F is isomorphic to Zp, for some prime p.

Added by Francisco Javier B.

Question

Please give Ace some feedback