Early work combining recursion theory and algebra had (at least) two different sets of motivations. First the precise setting of recursion theory offered a chance to make formal classical concerns as to the effective or algorithmic nature of algebraic constructions. As an added benefit the formalization gives one the opportunity of proving that certain constructions cannot be done effectively even when the original data is presented in a recursive way. One important example of this sort of approach is the work of Frohlich and Shepardson [1955] in field theory. Another motivation for the introduction of recursion theory to algebra is given by Rabin [1960]. One hopes to mathematically enrich algebra by the additional structure provided by the notion of computability much as topological structure enriches group theory. Another example of this sort is provided in Dekker [1969] and [1971] where the added structure is that of recursive equivalence types. (This particular structural view culminates in the monograph of Crossley and Nerode [1974].)

More recently there is the work of Metakides and Nerode [1975], [1977] which combines both approaches. Thus, for example, working with vector spaces they show in a very strong way that one cannot always effectively extend a given (even recursive) independent set to a basis for a (recursive) vector space.