# The smallest subring of $A$, and the characteristic of $A$ if $|A|$ is prime.

So basically I have two questions.

Let $A$ be a ring. What is the smallest subring of $A$.

My answer: the smallest subring of $A$ is of cardinal $Char(A)$.
My proof: Let $n = Char(A)$. Then $A$ has a sub-ring $B$ that is in bijection with $\mathbb{Z}/Char(A)\mathbb{Z}$. Thus the group $(B,+)$ is of cardinal $n$, and thus $|B| =n$. Suppose there exists another subring $C$ such that $|C| < |B|$. But $Char(C)=Char(A)$, thus $C$ has a subring $D$ that is isomorphic to $\mathbb{Z}/Char(A)\mathbb{Z}$, thus $|D|=n < |B| =n$. Which is absurd.

Then:

What is $Char(A)$ equal to if $|A|$ is prime?

My answer: $Char(A)=|A|$ (at least that's what I think, I am unsure).

My attempt: If $A$ is a ring, then it has a subring $S$ such that $Char(S)=Char(A)$. I have shown that $|S| = Char(A)$. As $(A,+)$ and $(S,+)$ are groups, we have $|S|$ divides $|A|$. Thus $|S|= 1 \text{ or } |A|$. And here I have difficulties to continue. Any ideas?

Your statements are almost correct, but you forgot one case: the smallest subring can be isomorphic to $\mathbb{Z}$.
For every (unital) ring $A$ there exists a unique ring homomorphism $\chi_A\colon\mathbb{Z}\to A$; in particular the image of $\chi_A$ is the smallest subring of $A$ (prove it).
Let $n\mathbb{Z}$ be the kernel of $\chi_A$, with $n\ge0$.
If $n=0$, then $\chi_A$ is injective, so the smallest subring of $A$ is isomorphic to $\mathbb{Z}$ (and the characteristic of $A$ is $0$). If $n>0$ the smallest subring of $A$ is isomorphic to $\mathbb{Z}/n\mathbb{Z}$ (and the characteristic of $A$ is $n$).
If $|A|$ is prime, then $A$ has no proper (unital) subring, because a subring is also an additive subgroup and Lagrange’s theorem applies.
Prove that there is only one ring with characteristic $0$, namely the zero ring. Hint: $0=1$ is both the multiplicative annihilator and the multiplicative identity.