Let $(R,\mathfrak{m})$ be a Noetherian local ring of Krull-dimension $d$ with maximal ideal $\mathfrak{m}$.
By definition a system of parameters are elements $a_1,\ldots, a_d\in \mathfrak{m}$ with $$ \mathfrak{m}=\sqrt{(a_1,\ldots, a_d)}. \hspace{5ex}\text{(I)} $$ One can prove, that such a system of parameters always exist.
On the other hand let $b_1,\ldots, b_n\in \mathfrak{m}$ be a minimal set of generators of $\mathfrak{m}$, ie $$ \mathfrak{m}=(b_1,\ldots, b_n). \hspace{8ex}\text{(II)} $$ and you cannot take away a $b_i$ without losing this property. By the principal ideal theorem, one has $n\geq d$ and $R$ is called regular if $n=d$.
I am confused about the connection between these properties. It seems that the ''only'' obstruction for $R$ to be regular is the radical-sign in (I). What is an example that this radical cannot be ignored when $R$ is a reduced ring? (certainly not every reduced $R$ is regular.) To be reduced or not is not a ''geometric property'' of a ring, meaning that you cannot see this after taking the associated variety $V$.
So my unfortunately very spongy question is: Could someone please help me to understand why the radical-sign is needed in (I), perhaps with a concrete reduced example? What is morally the reason that elements with (I) do always exist but (II) with $n=d$ is a very strong property? Thank you very much.
