Image of ideals under linear K-derivations and the LNED conjecture

Let K be a field of characteristic zero and K[X]=K[x<inf>1</inf>,x<inf>2</inf>,…,x<inf>n</inf>] be the polynomial algebra in n variables over K. We show that, for a linear K-derivation d of K[X] and the maximal ideal m=(x<inf>1</inf>,x<inf>2</inf>,…,x<inf>n</inf>) of K[X], if d(m) is a Mathieu-Zhao subspace of K[X], then the image of every m-primary ideal under d forms a Mathieu-Zhao subspace of K[X]. Additionally, we observe that the image of all monomial ideals under the K-derivation d=f∂<inf>x<inf>1</inf></inf> of K[X], for f∈K[X] forms an ideal of K[X]. Finally, we prove that the image of certain monomial ideals under a linear locally nilpotent K-derivation of K[x<inf>1</inf>,x<inf>2</inf>,x<inf>3</inf>] defined by d=x<inf>2</inf>∂<inf>x<inf>1</inf></inf>+x<inf>3</inf>∂<inf>x<inf>2</inf></inf> forms a Mathieu-Zhao subspace.