Subfields of Rings I am currently working through an undergraduate class in Galois Theory. I have come across a question that I am unsure about. 
Can a ring that is not a field, have a subring that satisfies the conditions to be a field? 
From some investigation, it appears that a ring with prime characteristic $p$ would have a subfield of the form $\{ a * 1_r \,| \,a \in \mathbb{Z}_p \}$. Is this true and are there other possible fields within rings?
 A: Let $k$ be a field. Then $k[t]$ is not a field but has $k$ as a subfield. If $p(t) \in k[t]$ is a polynomial that is not a unit or irreducible then $k[t]/(p(t))$ is not a field and also contains a copy of $k$. 
A: *

*Consider the ring $ {\mathbf{M}_{2}}(\mathbb{F}) $ of $ (2 \times 2) $-matrices with entries from a field $ \mathbb{F} $. The subring
$$
\left\{ \left[ \matrix{x & 0 \\ 0 & 0} \right] ~ \Bigg| ~ x \in \mathbb{F} \right\}
$$
is isomorphic to $ \mathbb{F} $.

*Consider the ring of $ \mathbb{F} $-valued functions on a set. The subring of constant functions is isomorphic to $ \mathbb{F} $.
A: 
Can a ring that is not a field, have a subring that satisfies the conditions to be a field?

Yes. Here's an infinite family of examples. Let $\phi : R_0 \longrightarrow R_1$ be a homomorphism of unital rings. Suppose $R_0$ is a field and $R_1$ is a nonzero ring.
It follows that $\phi$ is injective (proof omitted). Since $\phi$ is a ring homomorphism, $\phi(R_0)$ is a subring of $R_1$. Since $\phi$ is injective, $\phi(R_0)$ is isomorphicto $R_0$, which is a field. So, $\phi(R_0)$ is a subring of $R_1$ and also a field.
