I'm currently learning Abstract Algebra for the first time using Gallian. One question that I have asked myself more than once when doing problems for Chapter 15 (Ring Homomorphisms) is if there is a standard way to describe the 'smallest subring containing $a$' for a non-zero $a \in R$. (because the zero ring is just $\{0\}$.)
That is, in group theory, we can easily prove that the smallest subgroup containing non-identity element $a \in G$ is the cyclic group generated by $a$, $\langle a \rangle$, or in set-builder notation, $\{a^k | k \in \mathbb{Z}\}$. So we can confidently say that any element in this group will have the form $a^k$, and thus prove properties of this group.
But now, when working with rings and fields, the question becomes a lot less simple as the requirement of a well-defined 'multiplication' operation in any subgroup widens the amount of minimum elements 'generated' by some $a \in R$. The 'fields' version is marginally more difficult, as there needs to be inverses included as well.
To find the general form of an element in the smallest possible subring $R_a$ containing $a \in R$, I have found that the following set-builder definition:
$$R_a = \{ \sum^N_{1 = i} (n_i a)^{k_i} | N, {k_i} \in \mathbb{Z}^+, n_i \in \mathbb{Z} \}$$
Does define a valid subgroup, in that for some arbitrary elements $x, y \in R_a$ where $x = \sum^N_{1 = i} (n_i a)^{k_i}$ and $y = \sum^M_{1 = i} (m_i a)^{j_i}$, the elements $x-y$ and $xy$ both can be expanded out into the same form. I am also reasonably confident that, for any subring $S \subset R$ containing $a$, if some $x = \sum^N_{1 = i} (n_i a)^{k_i}$ is not contained in $S$ as well, that it can be shown inductively to contradict the subring test.
Similarly, my tentative definition for the smallest possible subfield containing some $a\in F$, $F_a$, is nearly identical, namely:
$$F_a = \{ \sum^N_{1 = i} (n_i a)^{k_i} | N \in \mathbb{Z}^+,k_i, n_i \in \mathbb{Z} \}$$
The only difference is that the power $k_i$ can now be negative (indicating a closure under inverses). I am also reasonably confident that this passes the subfield test.
Finally, to show why having notation like this is important, notice that it helps quite drastically in some proofs, i.e:
All Field Automorphisms act as the identity map on the prime subfield:
Proof: Notice that any automorphism $\phi$ maps the unity of $F$ to itself, namely $\phi(1) = 1$. Further, notice that for every element $x = \sum^N_{1 = i} (n_i 1)^{k_i}$ in the smallest possible ring containing 1 $R_1$,
$$\phi(x) = \phi(\sum^N_{1 = i} (n_i 1)^{k_i}) = \sum^N_{1 = i} \phi((n_i 1)^{k_i}) = \sum^N_{1 = i} ((n_i\phi(1))^{k_i}) = \sum^N_{1 = i} (n_i 1)^{k_i} = x$$
I.e any automorphism $\phi$ acts as the identity on the smallest possible ring containing the unity, $R_1$. Since the prime subfield is necessarily contained within $R_1$, every automorphism acts as the identity on the prime subfield as well. $\blacksquare$.
The thing is, I have not been able to find any standard set-builder definitions of both the smallest subring/subfield containing $a$ in either the text or the internet, so I was wondering if there was a more efficient, or better way to define these subrings.
Thanks!