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We first introduce and study the notion of semi-regular flat modules, and then show that a ring R is a strong Prüfer ring if and only if every submodule of a semi-regular flat R-module is semi-regular flat, if and only if every ideal of R is semi-regular flat, if and only if every R-module has a surjective semi-regular flat (pre)envelope.

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In this paper, we discuss some properties on Lucas modules. In details, we show that direct and inverse limits of Lucas modules are Lucas modules, and every R-module has a Lucas envelope and a Lucas cover. Moreover, some properties of direct and inverse limits of Lucas modules and some constructions and the unique mapping properties of Lucas envelopes and Lucas covers are investigated.