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In this paper, we investigate the notions of $\mathcal{X}^\bot$-projective,$\mathcal{X}$-injective and $\mathcal{X}$-flat modules and give somecharacterizations of these modules, where $\mathcal{X}$ is a class of left$R$-modules. We prove that the class of all $\mathcal{X}^\bot$-projectivemodules is Kaplansky. Further, if the class of all $\mathcal{X}$-projective$R$-modules is closed under direct limits, we show the existence of$\mathcal{X}^\bot$-projective covers and $\mathcal{X}$-injective envelopes overa $\mathcal{X}^\bot$-hereditary ring $R.$ Moreover, we decompose a$\mathcal{X}^\bot$-projective module into a projective and a coreduced$\mathcal{X}^\bot$-projective module over a self $\mathcal{X}$-injective and$\mathcal{X}^\bot$-hereditary ring. Finally, we prove that every module has a$\mathcal{W}$-injective precover over a coherent ring $R,$ where $\mathcal{W}$is the class of all pure projective modules.

Existence of covers and envelopes of a left orthogonal class and its right orthogonal class of modules