Question

Prove that there only one structure of ring with identity on abelian group z+

Answer 1

Answer:

Explanation:

Suppose (Z,+,∘,k) is a ring structure (k is the identity). Thus k∘x=x, for every x∈Z.

Suppose k>0. Then k=1+1+⋯+1, so

x=k∘x=1∘x+1∘x+⋯+1∘x=k(1∘x)

In particular, x∈kZ, for every x∈Z. Therefore k=1.

Then for m>0 and any n,

m∘n=(1+1+⋯+1)∘n=1∘n+1∘n+⋯+1∘n=n+n+⋯+n=mn

If m<0, m∘n=−((−m)∘n)=−(−mn)=mn.

Finish it up, by proving that k≤0 is impossible.