Is the absorbing property of semiring $S$ a necessary criteria for $S$ to be called semiring?

Question

An algebraic structure say $(S, +, 0, 1)$ is called semiring if $(S, +, 0)$ and $(S, ., 1)$ are monoid and multiplication distributes over addition from both left and right. It is also encountered that if $0$ is additively neutral element, then it is multiplicatively absorbing. The question is that whether the absorbing property necessary?. In my effort to construct a semiring i am faced with a problem that despite satisfying all other axioms, absorption property doesn't hold properly as i started with a neutral $0$ of lower dimensional and got higher dimensional $0$ after multiplying it with some element in $S$ as per my definition of the operation (.), i.e., $x.0=0 ~\forall ~x\in S$ here, $0$ in R.H.S is of higher dimension than that of L.H.S so, here i am really stucked if absorption property is a criteria for semirings.

Answer 1

According to wikipedia, it is a separate axiom and indeed necessary for a semiring. If $(S, +, 0)$ has the cancellation property, then $0x=0$ follows from the rest of the semiring axioms because of $$ 0x = 0x\\ (0+0)x = 0x\\ 0x + 0x = 0x\\ 0x = 0 $$ but the last step is not allowed in a general monoid so we need to state $0x = 0$ separately.