Let $R$ be a ring (commutative, with unit), $M$ an $R$-module, and $x_1, \dotsc, x_n \in R$. Consider the following two conditions:

- For all $i$, the homomorphism $$\frac{M}{(x_1, \dotsc, x_{i-1})M} \stackrel{x_i}{\longrightarrow} \frac{M}{(x_1, \dotsc, x_{i-1})M}$$ is injective.
- $$\frac{M}{(x_1, \dotsc, x_n) M} \neq 0.$$

Taken together, these conditions give the definition for the $x_i$ to form an $M$-regular sequence. However, it is sometimes useful to consider Condition 1 by itself. For instance, this comes up in Eisenbud, *Commutative Algebra*, Exercise 6.7 (page 174 in my copy). Eisenbud sort of, but not really, calls such a sequence an "(almost) regular sequence."

Is there a standard term (or, for that matter, any reasonable term with a not-too-obscure reference) for a sequence of elements of $R$ (possibly contained in some fixed ideal of $R$, especially if $R$ is local) that satisfy Condition 1, but not necessarily Condition 2?

(Other related notions of "not-quite-regular sequence" would also be of interest.)