Projections of monogenic algebras

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 give an account of all monogenic algebraic algebras for which a projective image of the radical is not equal to the radical of a projective image.