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Introduction.The main purpose of this paper is to draw attention to certain functors, exactly analogous to the functors "Tor" and "Ext" of Cartan-Eilenberg[2], but applicable to a module theory that is relativized with respect to a given subring of the basic ring of operators. In particular, we shall show how certain relative cohomology theories for groups, rings, and Lie algebras can be subsumed under the theory of the relative Ext functor, just as (in [2]) the ordinary cohomology theories have been subsumed under the theory of the ordinary Ext functor.Among the various relative cohomology groups that have been considered so far, some can be expressed in terms of the ordinary Ext functor; these have been studied systematically within the framework of general homological algebra by M. Auslander (to appear). A typical feature of these relative groups is that they appear naturally as terms of exact sequences whose other terms are the ordinary cohomology groups of the algebraic system in question, and of its given subsystem.There is, however, another type of relative cohomology theory whose groups are not so intimately linked to the ordinary cohomology groups and