An ideal $I$ of a semiring $S$ is said to be k-ideal if $a\in I$ and $x\in S$, and if either $a+x\in I$ or $x+a\in I$ then $x\in I$.
An ideal $I$ of a semiring $S$ is said to be $subtractive$ if $a\in I$ and $a+x\in I$ then $x\in I.$
My observation: If $S$ is an additively commutative semiring, then both the definition seems to be the same except that unlike in the later case, the former case, it is specified that $x\in S$. Can we treat these two definition as the same in case of additively and multiplicatively commutative semiring? Or, are there any major differences?