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

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