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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.

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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.

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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.

Note:

Requirement of the presence of Identity depends on your requirement based what theorem or statement you want to prove.

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