Centrally stable algebras

Abstract We define an algebra A to be centrally stable if, for every epimorphism φ from A to another algebra B, the center Z ( B ) of B is equal to φ ( Z ( A ) ) , the image of the center of A. After providing some examples and basic observations, we consider in somewhat greater detail central stability in tensor products of algebras, and finally establish our main result which states that a finite-dimensional unital algebra A over a perfect field F is centrally stable if and only if A is isomorphic to a direct product of algebras of the form C i ⊗ F i A i , where F i is a field extension of F, C i is a commutative F i -algebra, and A i is a central simple F i -algebra.