Remark 1

It follows from formulas (6) and (7) that S is secant for M if and only if it is secant for $A/\alpha$, where $\alpha$ is the annihilator of M. 2) Let S and S’ be two disjoint subsets of $m_A$. For $S \cup S'$ to be secant for M, it is necessary and sufficient that S be secant for M and S’ secant for $M' = M/SM$. This follows from inequality (8) and the formula

3) Let $x \in m_A$ and $n \geq 1$ be an integer. It is immediate that the modules $M/xM$ and $M/x^nM$ have the same support, hence the same dimension. Consequently, x is secant for M if and only if $x^n$ is secant for M. From this and Remark 2, one deduces at once the following result: let $x_1, ..., x_r$ be elements of $m_A$ and $n_1, ..., n_r$ integers $> 0$; then the sequence $(x_1, ..., x_r)$ is secant for M if and only if the sequence $(x_1^{n_1}, ..., x_r^{n_r})$ is secant for M.