Corner Ring Theory: A Generalization of Peirce Decompositions, I

In this paper, we introduce a general theory of corner rings in noncommutative rings that generalizes the classical notion of Peirce decompositions with respect to idempotents. Two basic types of corners are the Peirce corners eRe (e2 = e) and the unital corners (corners containing the identity of R). A general corner is both a unital corner of a Peirce corner, and a Peirce corner of a unital corner. The simple axioms for corners engender good functorial properties, and make possible a broader study of subrings with only some of the features of Peirce corners. In this setting, useful notions such as rigid corners, split corners, and semisplit corners also come to light. This paper develops the foundations of such a corner ring theory, with a view toward a unified treatment of various descent-type problems in ring theory in its sequel.