# Subring criteria

To check whether the set $$S$$ is a subring of the Ring $$R$$, you check for closure under addition and multiplication but do you check for an additive or multiplicative inverse. In my book, for subrings of Rings like $$\mathbb{R}$$ it checks for multiplicative inverse but to check for example if $$\mathbb{Z[i]}$$ is a subring of $$\mathbb{C}$$ it checks for the additive inverse.

Yes, one must check the existence of an additive inverse. Of course, since we are working with a subset $S$ of a ring $R$, all we need to check is whether or not $s\in S\implies-s\in S$. On the other hand, there is no need to check the existence of multiplicatives inverses, since their existence is not required.
A Subset $$S$$ of ring $$R$$ is said to a subring of $$R$$ if the following conditions hold:
(i) $$S$$ is closed under the subtraction.
(ii) $$S$$ is closed Under Multiplication.