Coproducts and some universal ring constructions

Let R be an algebra over a field k, and $P,Q$ be two nonzero finitely generated projective R-modules. By adjoining further generators and relations to R, one can obtain an extension S of R having a universal isomorphism of modules, i:P ⊗ RS Q ⊗ RS. We here study this and several similar constuctions, including (given a single finitely generated projective R-module P) the extension S of R having a universal idempotent module-endomorphism e:P ⊗ S → P ⊗ S, and (given a positive integer n) the k-algebra S with a universal k-algebra homomorphism of R into its n × n matrix ring, f:R → mₙ(S). A trick involving matrix rings allows us to reduce the study of each of these constructions to that of a coproduct of rings over a semisimple ring R₀ (( = k × k × k,k × k, and k respectively in the above cases), and hence to apply the theory of such coproducts. As in that theory, we find that the homological properties of the construction are extremely good: The global dimension of S is the same as that of R unless this is 0, in which case it can increase to 1, and the semigroup of isomorphism classes of finitely generated projective modules is changed only in the obvious fashion; e.g., in the first case mentioned, by the adjunction of the relation $[P] = [Q]$. These results allow one to construct a large number of unusual examples. We discuss the problem of obtaining similar results for some related constructions: the adjunction to R of a universal inverse to a given homomorphism of finitely generated projective modules, f:P → Q, and the formation of the factor-ring R/TP by the trace ideal of a given finitely generated projective R-module P (in other words, setting $P = 0$). The idea for a category-theoretic generalization of the ideas of the paper is also sketched.