Subfields of Rings

Question

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?

Like a finite-dimensional central division algebra over a field? — awllower, Feb 19 '13 at 03:00
For example, Ring of quaternion numbers is not a field. But it has contain subfield $\mathbb{R}$ and $\mathbb{C}$. — Hanul Jeon, Feb 19 '13 at 03:00
Any field F is a subring of the polynomial ring F[x], which is not a field. — Math Gems, Feb 19 '13 at 03:01

score 9 · Accepted Answer · edited Feb 19 '13 at 03:05

9

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 $k$.

edited Feb 19 '13 at 03:05

hardmath

37,015

answered Feb 19 '13 at 03:01

JSchlather

15,427

score 1 · Answer 2 · answered Feb 19 '13 at 04:52

1

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} $.

answered Feb 19 '13 at 04:52

Haskell Curry

19,524

2

For your first example, you have to be a little careful. Because some authors require a subring to have the the same unit as the ring. – JSchlather Feb 19 '13 at 08:19
Sure, I know that. I’m not assuming that subrings must inherit the same unit as the parent ring. For my first example, the subring has a meaningful ring unit that is distinct from the parent ring’s own unit, which is the identity matrix. – Haskell Curry Feb 20 '13 at 00:38

score 0 · Answer 3 · answered Nov 09 '17 at 05:36

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.

Subfields of Rings

3 Answers3