In the context of category theory and module theory, a **projective cover** is a particular type of object that serves as a "minimal" projective object that maps onto a given object (or module) in a way that reflects certain structural properties.
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A projective cover of a module is an essential surjection from a projective module. For a finite-dimensional algebra, every finite-dimensional module has one; the projective cover of a simple module is indecomposable and has that simple module as its head.