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Let $k$ be a field of characteristic zero and $R$ a $k$-algebra. In thispaper we study homogeneous $R$-lnds $D$ on $R[X,Y,Z]$ with respect to thestandard weights $(1,1,1)$. We show that when $R$ is a PID, $rank(D)$ can be atmost $2$ if $\deg(D) \leqslant 3$. As a consequence we obtain a certain classof homogeneous lnds on $k^{[4]}$ whose kernel is $k^{[3]}$. Further when $R$ isa Dedekind domain, we give a bound for minimum number of generators of$\ker(D)$ as an $R$-algebra if $\deg(D) \leqslant 3$.

Some results on homogeneous locally nilpotent $R$-derivations on $R[X,Y,Z]$