# Smallest subring of $\mathbb Q$ containing $3/10$

Let $R$ be the smallest subring of $\mathbb Q$ (the field of rational numbers) that contains $3/10$ ($R$ doesn't have to be a unital ring). Does $1 \in R$?

Is the desired smallest subring this one: $R_1=\{\frac{3a}{10^b},a,b \in \mathbb Z \}$ or this one: $R_2=\{\frac{3^ca}{10^b},a,b,c \in \mathbb Z \}$? Either case, $1 \notin R_i$, because $3\nmid10^b$, correct?

• Isn't $R_1=R_2$ since you can put powers of $3$ into $a$? – Cheerful Parsnip Feb 24 '17 at 20:11
• I was wondering the same thing! – John Feb 24 '17 at 20:14
• Oh wait, actually they are not the same. $R_2$ allows $c=0$, giving $a/10^b$. – Cheerful Parsnip Feb 24 '17 at 20:18

Your $R_1$ is not a ring because it is not closed under multiplication. Your $R_2$ contains $1$ (for $a=1, b=c=0$). However, if you modify $R_2$ to $\{\frac{3a}{10^c}\mid a\in\Bbb Z, c\in\Bbb N\,\}$, you are fine. Even if this should turn out not to be the smallest such ring, it is sufficient to know that it is a sub-ring containing $\frac3{10}$ and not containing $1$; the smallest can only contain less elements.
• Why is $\mathbb R_1$ not closed under multiplication? – Henning Makholm Feb 24 '17 at 19:59
• Actually, if my $R_1$ is not a ring, then your $R_2$, shouldn't be also not a ring? – John Feb 24 '17 at 20:11
• Like you, I originally thought that the minimal ring needs $c\in\mathbb N$, not $c\in\mathbb Z$. But in fact that is an illusion $-$ they define the same object. – TonyK Feb 24 '17 at 20:24