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For any linear inequality in three variables $\mathcal{L}$, we determine (ifit exist) the smallest integer $R(\mathcal{L}, \mathbb{Z}/3\mathbb{Z})$ suchthat: for every mapping $\chi :[1,n] \to \{0,1,2\}$, with $n\geq R(\mathcal{L},\mathbb{Z}/3\mathbb{Z})$, there is a solution $(x_1,x_2,x_3)\in [1,n]^3$ of$\mathcal{L}$ with $\chi(x_1)+\chi(x_2)+\chi(x_3)\equiv 0$ (mod $3$). Moreover,we prove that $R(\mathcal{L}, \mathbb{Z}/3\mathbb{Z})=R(\mathcal{L}, 2)$, where$R(\mathcal{L}, 2)$ denotes the classical $2$-color Rado number, that is, thesmallest integer (provided it exist) such that for every $2$-coloring of$[1,n]$, with $n\geq R(\mathcal{L}, 2)$, there exist a monochromatic solutionof $\mathcal{L}$. Thus, we get an Erd\H{o}s-Ginzburg-Ziv type generalizationfor all lineal inequalities in three variables having a solution in thepositive integers. We also show a number of families of linear equations inthree variables $\mathcal{L}$ such that they do not admit suchErd\H{o}s-Ginzburg-Ziv type generalization, named $R(\mathcal{L},\mathbb{Z}/3\mathbb{Z})\neq R(\mathcal{L}, 2)$. At the end of this paper somequestions are proposed.

Erd\H{o}s-Ginzburg-Ziv type generalizations for linear equations and linear inequalities in three variables