Enveloping ringoids

We present notions of module over a universal algebra, and linear representation of a universal algebra, which have gained currency with categorical algebraists, we give several intriguing examples of these objects, showing that important aspects of universal algebras can be studied in this context, and we describe the theory ofenveloping ringoids, which have categories of left modules equivalent to corresponding categories of modules and linear representations. Algebras in many of the familiar varieties of algebras, which have underlying groups, turn out to have enveloping ringoids that are equivalent to familiar rings. Nonempty algebras in an abelian varietyV have an enveloping ringoid equivalent toR(V), the so-calledring of the variety V.Dedicated to the memory of Alan DayPresented by J. Sichler.