On an open question concerning idempotent matrices over GCD domains

The classification of idempotent matrices up to similarity is a foundational problem in matrix theory, serving as a bridge between the algebraic properties of rings and the geometric nature of their associated projective modules. A primary objective in this field is to identify the broadest classes of rings, specifically commutative domains, where every idempotent matrix is similar to a standard diagonal idempotent [Formula: see text]. Very recently, Călugăreanu and Lee [Carpathian J. Math. 42 (2026), 613–623] posed the open question of whether this property holds for all commutative GCD domains. While their work established this diagonal reduction for [Formula: see text] idempotents, we give a definitive negative answer for the general case. Let [Formula: see text] be the real affine coordinate ring of the two-sphere. We prove, by Nagata’s criterion, that [Formula: see text] is a unique factorization domain and hence a GCD domain. We explicitly construct an idempotent matrix [Formula: see text] in [Formula: see text]. Its image is the algebraic tangent module of the sphere, which is a finitely generated projective [Formula: see text]-module of rank [Formula: see text]. This module is not free, because freeness would imply the existence of a nowhere-zero continuous tangent vector field on [Formula: see text], contradicting the Hairy Ball Theorem. Therefore, [Formula: see text] is not similar over [Formula: see text] to [Formula: see text], nor to any standard diagonal idempotent. By adjoining zero blocks, this yields counterexamples for all sizes [Formula: see text], thereby resolving the open question in the negative for every [Formula: see text] and showing that the stated [Formula: see text] diagonalization theorem cannot hold under the stated hypotheses. The argument pinpoints the exact module-theoretic obstruction: over domains, diagonal similarity for all idempotents is equivalent to projective-freeness of the coefficient ring, a condition not implied by the GCD property.