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Let $I$ an ideal of $\mathbb{C}[X_1, \ldots X_n]$ and define the arithmetical rank of $I$ as: $$ ark(I) = \textrm{min} \left\{m \in \mathbb{N}, \exists f_1, \ldots, f_m \in \mathbb{C}[X_1, \ldots X_n]...

I expect this to be true and proven, but I can't find any proofs of this. So anyone can confirm or deny this? Let $R$ be a commutative ring, and let $M$ be a $kn\times kn$ matrix, which can be split ...

This is a cross-post from my original question at math.se. I decided to post here because it seems more difficult than I originally thought. Let $R=\mathbb C[x_1,\ldots,x_r]$ be a polynomial ring. ...

Consider a 2x2 matrix $A$ with entries from $\mathbb{C}[x,y]$. Assume that $\mathrm{det} A$ is a square. Is it true that then $A$ can be represented as a noncomuting product $A=A_1 A_2 … A_{2n}$, in ...

Given a $4 \times 4$ matrix $S$ over a commutative ring $R$. I want to consider it as a $2\times 2$ matrix over $M_2(R)$. Lets say $S=\left(\begin{array}{cc} A&B \\\ C&D\end{array}\right)$ ...

Suppose $A\in R^{n\times n}$, where $R$ is a commutative ring. Let $p_i \in R$ be the coefficients of the characteristic polynomial of $A$: $\operatorname{det}(A-xI) = p_0 + p_1x + \dots + p_n x^n$. I ...