Prelim problems 20 & 22 J.O

**20. (Preliminary for T, March 20; finalized Sat, March 24) Prove that the polynomial ring Z[x] over the integers is countably infinite.**

Well the this ring has one to one correspondence to the integers which has one to one correspondence to the natural numbers. Thus is countably infinite

**22. (Preliminary for T, March 20; finalized Sat, March 24) Let (R,0,1,+,⋅,≤) be an ordered ring with identity.**

**a. Prove that if x>0, then −x<0.
b. Prove that 1>0.**

x+(-x)>0+(-x)

0> -x

Suppose x>0

x(1/x) > 0(1/x)

1>0

page revision: 1, last edited: 20 Mar 2018 05:02