The first isomorphism theorem is usually stated in the following form:
Let $f:R\rightarrow S$ be a surjective homomorphism of rings with kernel $K$. Then the quotient ring $R/K$ is isomorphic to $S$, and $K = kernel f$.
In the proof of the theorem, there usually appears a function such as $\phi(a + K)=f(a)$. I am trying to apply the first isomorphism theorem to solve the following problem by imitating the strategy from its proofs. I have to show that $Z_{20}/(5)\cong Z_{5}$. So I let $f:Z_{20}\rightarrow Z_{5}$ be defined by $f(a+Z_{20})=a+Z_{5}$ and since $Im(f)=a+Z_{5}$ and $kernel f =\{[0]_5, [5]_5, [10]_5, [15]_5\}$. At this point, do I just define the function $\phi(a+ ker f)=Im(f)=a+ Z_{5}$. Is this the proper way going about the solution? The reason I am asking is because in many (not all) instances where i see the application of first isomorphism theorem is use. I see an explicit function $f$ being written out mapping from $R$ to $S$ or to $Im(f)$ being set up, then the kernel is found of $f$ is described and then by the isomorphism theorem, it is solved. The solutions usually skips the step of writing out explicitly the the function $\phi(R/K) = Im(f)$ Also, I am just wondering why that is. Thank you in advance
Best Answer
The “gold standard” way to apply the First Isomorphism Theorem to prove that a quotient $R/I$ is isomorphic to $S$ is to first define a surjective morphism $f\colon R\to S$, and then prove that $\mathrm{ker}(f)=I$. Then the First Isomorphism Theorem does all the work for you. The reason you never see the map $\Phi$ in the applications is that you don’t want to go through the every step of the proof each time you want to apply the theorem. That’s the whole point of proving the theorem in the first place!
Your notation leaves something to be desired. The elements of $\mathbb{Z}_{20}$ do not look like $a+\mathbb{Z}_{20}$: that doesn’t even make sense, as you never write an element of a ring $R$ as $a+R$. I’m going to guess that you define $\mathbb{Z}_n$ as the quotient $\mathbb{Z}/n\mathbb{Z}$, so that elements of $\mathbb{Z}_{20}$ are of the form $a+20\mathbb{Z}$ with $a\in\mathbb{Z}$, and the elements of $\mathbb{Z}_5$ are of the form $b+5\mathbb{Z}$, with $b\in\mathbb{Z}$.
So, to prove that $\mathbb{Z}_{20}/\langle 5+20\mathbb{Z}\rangle$ is isomorphic to $\mathbb{Z}_5$, we want to define a morphism $\mathbb{Z}_{20}\to\mathbb{Z}_5$ whose kernel will be exactly $\langle 5+20\mathbb{Z}\rangle$.
Your definition works. But you need to show that it is a ring homomorphism in order to use the Theorem. Once you prove it is a ring homomorphism, the fact that the kernel is precisely $$\{0+20\mathbb{Z}$, 5+20\mathbb{Z}, 10+20\mathbb{Z},15+20\mathbb{Z}\} = \langle 5+20\mathbb{Z}\rangle$$ gives you the desired isomorphism, via the Isomorphism Theorem. The isomorphism is going to be annoying to write, since the elements of the quotient group are cosets of cosets... It would look something like $$\Phi\Bigl((a+20\mathbb{Z})+\langle 5+20\mathbb{Z}\rangle\Bigr) = a+5\mathbb{Z}.$$