The semiring $S$ is an algebraic structure with two binary operations $\oplus$ and $\odot$, such that $(S, \oplus)$ is an Abelian monoid with identity $0_{S}$, $(S, \odot)$ is a monoid with identity $1_{S}$, multiplication is distributive over addition and $0_{S}$ is an absorbing element for multiplication. However, there might not exist inverse elements for $\oplus$ or for $\odot$. In this work we present semirings, their properties, various examples and some examples of applications, in particular in optimization. We concentrate on the semirings of matrices. We define different ranks of matrices over semirings, which in general do not coincide. We know that there are many equivalent definitions of ranks of matrices over the field. These equivalences need not hold for matrix ranks over the semiring. In this work we prove some inequalities among ranks and show by examples that the strict inequalities might apply. We also prove some inequalities that hold for the rank of the sums of two matrices and for the rank of the product of two matrices.
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