Three sets of conditions on rings

We define a set of conditions ${\mathfrak {L}_m}$ on a ring $R$ using the notion of $R$-dependence of elements. We prove that ${\mathfrak {L}_1},{\mathfrak {L}_2}, \cdots$ is a strictly decreasing sequence of conditions. Two other sequences of conditions are considered and we prove that they are also strictly decreasing and we obtain their relation to ${\mathfrak {L}_m}$.