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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:

  1. 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.
  2. $$\frac{M}{(x_1, \dotsc, x_n) M} \neq 0.$$

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.)

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I will just note that the term "almost regular has been taken":… – Hailong Dao Jul 31 '11 at 22:57

A reasonable term with a not-too-obscure reference: that is called a weak $M$-sequence by Bruns and Herzog in Cohen-Macaulay Rings (p. 3 in the edition I have).

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This is certainly a valid reference (+1). The trouble is, based on google searches, that "weak M-sequence" is more commonly used to refer to a sequence in which condition (i) is replaced by the condition that everything annihilating $x_i$ annihilates your entire ideal (e.g., your maximal ideal, if $R$ is local). – Charles Staats Jul 31 '11 at 2:43

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