I am currently reading a proof of the fact that every regular Noetherian local ring $R$ is an integral domain.
The proof argues by induction on $d=\operatorname{dim}R$. The base case $d=0$ is clear. Now let $d>0$ and let $x_1,\ldots,x_d$ be a system of parameters generating the unique maximal ideal $m$ and let $P_1,\ldots,P_r$ denote the minimal prime ideals in $R$. By prime avoidance, choose an element $a\in m\setminus m^2$ which is not contained in any of the minimal prime ideals. Then, since $a,x_2,\ldots,x_d$ form a system of parameters of $R$, the dimension of $R/(a)$ is $d-1$. Furthermore, $x_2,\ldots,x_d$ give rise to a system of parameters of $R/(a)$ generating the maximal of $R/(a)$. Therefore $R/(a)$ is regular and by the induction hypothesis, an integral domain. In particular $(a)$ is prime in $R$ and therefore there exists a minimal prime, say $P_1$, such that $P_1\subseteq (a)$. It follows $P_1=(a)P_1$ and by Nakayma's Lemma $P_1=0$, implying $R$ is an integral domain.
What I don't see is, where did we need, that $a\notin m^2$? What goes wrong if $a$ happens to be in $m^2$?
EDIT: The Element $a$ has the form $$ a=x_1+\sum_{i=2}^d c_i x_i $$ for some coefficients $c_i\in R$.
EDIT2: I just read in Rotman's "Advanced Modern Algebra" Proposition 11.165, which states that elements $x_1,\ldots, x_d$ form a minimal generating set for $m$ if and only if the cosets $\overline{x_i} = x_i + m^2$ form a basis of $m/m^2$. So apparently not lying in $m^2$ is necessary for being a minimal system of parameters for a regular Noetherian local ring, since otherwise they would be linearly dependent in $m/m^2$. However, I don't understand why $m(m/B)=(B+m^2)/B$ holds in Rotman's proof. So if we could settle this, I would be happy.
That is just the coset module action and coset arithmetic. By definition, $r(s+B)=rs+B$, and so $m(m/B)$ is some submodule of $m/B$ which contains everything of the form $m_1m_2+B$ for every $m_1, m_2\in m$.
One would like to say this is $m^2/B$ but there is no guarantee $m^2$ contains $B$. Therefore the best we can say is that it is $(m^2+B)/B$