Can somebody please look at my work and critique. Thanks in advance!
Let p be a prime number, let $\mathbb{Q}$ be the field of rational numbers, and define the set:
$$S = \{{n/p^e \,|\, n \in \mathbb{Z}, e \in \mathbb{Z}}\} \subset \mathbb{Q}$$
Show that S is a subring of $\mathbb{Q}$.
First we show that ($S,+$) is a subgroup of ($\mathbb{Q},+$):
$n/p^e + n/p^e = 2n/p^e \in S \qquad$ closed under addition
$n/p^e + 0 = n/p^e \in S \qquad$additive identity
$n/p^e + \left(-n/p^e\right) = 0 \in S \qquad$ Inverse
Hence, ($S,+$) is a subgroup of ($\mathbb{Q},+$)
Next we show ($S,\times$) is closed and $1 \in S$:
$\left(n/p^e\right) \left(n/p^e\right) \in S \qquad$ closed under multiplication
$\left(n/p^e\right) \left(n/p^e\right)^{-1} = 1 \in S$
Hence, S is a subring of $\mathbb{Q}$
Best Answer
Alternatively, consider the map $\mathbb Z[x] \to \mathbb Q$ induced by $x \mapsto \frac 1p$.
This map is a ring homomorphism whose image is $S$. Therefore, $S$ is a subring of $\mathbb Q$.