# Properties of sequence of general elements

Let $I=(a_1,\ldots,a_n)$ and write $x_i=\sum_{j=1}^{n}{\lambda_{ij}}{a_j}$ for $1\leq i\leq s$ and $(\lambda_{ij})\in R^{sn}$. The elements $x_1,\ldots,x_s$ form a sequence of general elements in $I$ or $x_1,\ldots,x_s$ are general in $I$ if there exists a dense open subset $U$ of $k^{sn}$ such that the image $\overline{(\lambda_{ij})}\in U.$

suppose $x_1,\ldots,x_s$ form a sequence of general elements in $I.$ My questions are

1) $x_1,\ldots,x_s$ is a superficial sequence for $I$?

2) ${x_1}^*,\ldots,{x_s}^*$ is a filter regular sequence for $G(I)_+$ (where $G(I)=\oplus_{n\geq 0}\frac{I^n}{I^{n+1}}$ and $G(I)_+=\oplus_{n\geq 1}\frac{I^n}{I^{n+1}}$) ?

3) $x_i$ is regular element?

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Can you remind me what a superficial sequence is? Why would you think that $x_i$ is a regular element, this surely must depend on $R$ and $I$, right? Is $R$ a particular ring? –  Karl Schwede Aug 27 at 12:04