# Do I have to prove the 10 axioms for rings when proving that $R_0$ is a subring of $R$.

Let $R$ be a ring and $R_0$ a non-empty subset of $R$. Show that $R_0$ is a subring iff, for any $a,b∈R_0$, we have $a-b$, and $ab$ in $R_0$.

In the previous question, when proving that if for any $a,b∈R_0$, we have $a-b$, and $ab$ in $R_0$ then $R_0$ is a subring, do I have to individually prove that all of the axioms of closure, associativity, commutativity, identity for addition and multiplication, additive inverses and distributivity hold?

• The short answer is yes. The long answer is that most of these properties already hold (e.g. associativity, commutativity, distributivity) because they are true for elements of $R$ and $R_0$ consists of elements of $R$! – Eoin Feb 13 '17 at 7:42
• Thank you! I feel stupid now haha. – The Bosco Feb 13 '17 at 7:46
• Can I just say that because $ab$, $a$, and $b \in R_0$, if $ab=a\in R_0$ then $\exists 1$ such that $a \cdot 1 = a \in R_0$? – The Bosco Feb 13 '17 at 8:04
• One thing you will have to prove is that $0 \in R_{0}$, but this is straightforward. But there is no way you can prove that if $R$ has a unity $1$, then $1 \in R_{0}$. For instance, if $R$ is the ring of integers, then the even integers $R_{0}$ satisfy the two conditions, but of course $1 \notin R_{0}$. – Andreas Caranti Feb 13 '17 at 8:17
• But isn't the multiplicative identity a necessary axiom? – The Bosco Feb 13 '17 at 8:22