A Note on Idempotent Matrices: The Poset Structure and The Construction

Abstract

Idempotent elements play a fundamental role in ring theory, as they encode significant information about the underlying algebraic structure. In this paper, we study idempotent matrices from two perspectives. First, we analyze the partially ordered set of idempotents in matrix rings over a division ring. We characterize the partial order relation explicitly in terms of block decompositions of idempotent matrices. Second, over principal ideal domains, we establish an equivalent condition for a matrix to be idempotent, derived from matrix factorizations using the Smith normal form. We also consider extensions over unique factorization domains and constructions via the Kronecker product and the anti-transpose. Together, these results clarify both the structural and constructive aspects of idempotents in matrix rings. Moreover, the set of idempotent matrices over a field can be viewed as an affine algebraic variety.