Classification of Rings

Question

I am trying to classify rings of order 10. I believe the only possible ring is $\mathbb{Z_{10}}$. Thus I am trying to find a map from my ring $R$ to $\mathbb{Z_{10}}$. The most obvious map $f: R \rightarrow \mathbb{Z_{10}}$ such that $f(1)=1$ and $f(n)=[n]_{10}$. I was able to prove that it was a homomorphism and surjective. However, I am stuck at proving injectivity. Can anyone please give me some hints

@BrianFitzpatrick: Thanks for pointing it out. I am just having trouble with one subpart of the question which I believe is not mentioned in the above link. I would just want to know whether my approach is correct and get some hints. — Rutherford Mark, Feb 26 '14 at 06:27
Note that for a mapping of one set of ten elements to another set of ten elements to be surjective (onto), it must also be injective (1-1) by finiteness. — hardmath, Feb 26 '14 at 11:14

score 3 · Answer 1 · answered Feb 26 '14 at 05:23

3

Recall that the additive part of the ring must be an abelian group. What do you know about groups of order 10?

answered Feb 26 '14 at 05:23

user127296

92

We know they are cyclic groups. – Rutherford Mark Feb 26 '14 at 06:24

score 1 · Accepted Answer · answered Feb 26 '14 at 07:25

A surjection between two finite sets of the same size is injective, which finishes your proof. This has nothing to do with ring theory, it's simply a theorem for finite sets. (If your map satisfies $f(x) = f(y)$ for $x \neq y$, then $10 = |f(R)| < |R| = 10$.)

Hope that helps,

Classification of Rings

2 Answers2