Is every monosemiring an idempotent semiring?

To make my question clear, let me give definitions as follows: A semiring $(R, +, .)$ is said to be monosemiring if $x.y= x+y$ for all $x, y$ in $R$. And idempotent semiring is the one in which $x.x= x=x+x$ for all $x$ in $R$. From this definition, we don't see that every monosemiring is an idempotent too. But there are some authors who define that monosemiring in an idempotent semiring satisfying the identity $x.y=x+y$ for all $x,y$ in $R$. So, from this definition, it seems like every monosemiring is an idempotent too.

youshould define what you mean by monosemiring. (Unless you want to ask what the customary definition or notation is. But it is my impression that these algebraic structures are not very popular.) $\endgroup$ – Goldstern Nov 5 '17 at 14:54