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In this paper, we prove first that ($\\mathcal{F}_{n},\\mathcal{L}_{n}$) is a complete hereditary cotorsion theory, where $\\mathcal{F}_n$ (resp. $\\mathcal{L}_n$) denotes the class of all $R$-modules with flat dimension at most $n$ (resp. $L_{n}$-injective $R$-modules). Then we introduce the $L_{n}$-injective dimension of a module and $L_n$-global dimension of a ring. 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