Define the ideals of a semiring as follows:
A non empty subset $I$ of a semiring $S$ is said to be a left (resp. right) ideal of $S$ if: (1) $a+b\in I$ for all $a, b\in I$ and (2) $sa$ (resp. $as$) $\in I$ for any $a\in I$ and $s\in S$.
I consider that that the same definition holds true for an ideal $J$ of a ring $R$ and additionally $J$ needs to satisfy that $a-b\in J$ for all $a, b\in J$(this is needed since $R$ possesses additive inverse).
Conclusion: Every ideal of a ring is also an ideal of a semiring but the converse need not be true.
Please correct me if i am wrong.
Given a ring $R$ with unity $1$, any subset $I$ that contains zero and is closed under addition (hence an additive submonoid) and multiplication by arbitrary elements of $R$ on either side is in fact an additive subgroup (hence an ideal), because for any $i \in I$, $-i=(-1)i=i(-1)$, showing that $-i \in I$.