Submanifolds of the Euclidean spaces satisfying equality in the basic Chen's inequality have, as is known, many interesting properties. In this paper, we discuss on such submanifolds the curvature conditions of the form $E_2\cdot F_4=0$, where $E_2$ is the Ricci or the Einstein curvature operator, $F_4$ is any of the standard curvature operators $R, Z, P, K, C$, and $E_2$ acts on $F_4$ as a derivation.