Proposition 3

Let A be a noetherian local ring and M a nonzero finite a-module. For an element x of $m_A$ to be secant for M, it is necessary and sufficient that it belong to none of the minimal elements $p$ of $\mathrm{Supp}(M)$ such that $\dim(A/p) = \dim_A(M)$, and it is sufficient that the homothety $x_M$ of ratio x in M be injective. Let $\alpha$ be the annihilator of M. To say that x is secant for M means that x is secant for $A/\alpha$; the support of M consists of the prime ideals $p$ of A such that $\alpha \subset p$; and if $x_M$ is injective, the image of x in $A/\alpha$ is not a divisor of 0 in $A/\alpha$. Proposition 3 then follows from Corollary 2 of Proposition 2 of No. 1 applied to the ring $A/\alpha$.