Projections of Monogenic Algebras

Abstract

Let A and B be associative algebras treated over a same field F. We say that the algebras A and B are lattice isomorphic if their subalgebra lattices L(A) and L(B) are isomorphic. An isomorphism of the lattice L(A) onto the lattice L(B) is called a projection of the algebra A onto the algebra B. The algebra B is called a projective image of the algebra A. We give a description of projective images of monogenic algebraic algebras. The description, in particular, implies that the monogeneity of algebraic algebras treated over a field of characteristic 0 is preserved under projections. Also we provide a characterization of all monogenic algebraic algebras for which a projective image of the radical is not equal to the radical of a projective image.