I'm readind the article: A formula for the core of an ideal, by Bernd Ulrich and Claudia Polini and I'm having trouble each time the doubt involves the concept of general element of an ideal
Well, this isn't a concept widely known at the Commutative Algebra literature. The definition that was passed me is
Let $(R, \mathfrak{m})$ be a local ring and $I$ be an ideal of $R$. We say $x \in I$ is general element if $x$ is in non-empty Zariski open of $I/(\mathfrak{m}I)$.
The problem is that this definition seems complicated to manipulate and generate useful results. For example, at Lemma 4.2, the author says:
Let $R$ be Cohen-Macaulay local ring and $I$ be an ideal de $R$. If $ht(I)>0$ and $x \in I$ is general element, then $x$ is $R$-regular element.
Can someone help me at last resul I mentioned? Thanks very much!