I was studying an algebra book by Dummit and Foote, there I saw an example of subrings; $\mathbb H_{\mathbb Z}=$ $\{a+bi+cj+dk: a,b,c,d\in\mathbb Z\}$ is a subring of $\mathbb H_{\mathbb R}$=$\{a+bi+cj+dk:a,b,c,d\in\mathbb R\}$. So I was thinking if $\mathbb C=\{a+bi+0j+0k:a,b\in\mathbb R\}$ can be a subring of ring of quaternion $\mathbb H_{\mathbb R}$.
First I checked that this is the subset of $\mathbb H_{\mathbb R}$ which it is and it's also a ring in itself but I am having doubts about it. As we can see that the ring $\mathbb C$ is a commutative ring (in fact, a field), and $\mathbb H_{\mathbb R}$ is a noncommutative ring. So is this possible that a subring is commutative and the ring is not? Or I am getting everything incorrectly?
Please correct me if I am wrong. Any hints will be appreciated.
Best Answer
Of course it is possible for a noncommutative ring to have a commutative subring. In fact, every noncommutative ring has at least one commutative subring, namely the trivial ring.*
What is impossible is for a commutative ring to have a noncommutative subring: if $ab=ba$ for all $a,b\in R$, then the same must be true for any subset of $R$.
*The definitions of ring and subring vary from author to author. While what I said is true for the definitions given in Dummit and Foote, other sources (e.g. Wikipedia) disagree.