We know that the identity of a group and its sub groups are same. But I think in case of semirings, the additive identity of a sub semiring may be distinct from that of the semiring. I see a counter example in this regards. Please suggest me whether i am correct?
Edited.
I am thinking of the following counter example.
Set $S$ of all simple undirected graphs is a semiring under graph union (addition) and graph intersection (multiplication). Empty graph (vertex less graph) denoted by $(\emptyset, \emptyset)$ is the additive identity. Suppose $S'$ is the set of all those subgraphs of a graph $G\in S$ whose vertex set is $V(G)$. Then $S'$ is a sub semiring of $S$ under the same operations with additive identity $(V(G), \emptyset)$, which we call discrete graph.