hi
let $R$ be a noetherian ring and $I=(a,b,c,d)$ an ideal of R such that $grade(I,R) \geq 2$. How can we assume that $a ,b , c,d$ such that $a,b$ form a regular sequence and $c,d$ form a regular sequence? in fact: How can we replace the generators $a,b,c,d$ such that the new generators $e,f,g,h$ have the this property: $e,f$ form a regular sequence and $g,h$ form a regular sequence.
I think that there exist $r\in R$ such that $a+rb \notin ZD(R)$ and hence we can replace $a$ with $a+rb$ to assume that $a$ is regular, and continue in this way to replace the other generators. But i can not prove the existence of $r\in R$ with this property.
we also have the following lemma:
let $(R , m)$ be noetherian local ring and $P_1 , ... , P_n$ are prime but not maximal ideals, then there exist an infinite sequence of elements $r_i$ , $1\leq i \leq \infty$ ,such that for all $i\neq j$ ,$r_i - r_j \notin \cup P_i$. See Lemma 6.3 of "On The Support of local cohomology, T.Marley , Huneke ". But i can not prove this lemma for an arbitrary noetherian ring.
