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Note that for any such "closure operations" one has $cl(A) = cl(\langle A\rangle)$ where $\langle A\rangle$ denotes the ideal generated by $A$. Hence it can be defined as an operation on ideals. Now …

The short answer is "because you are considering $\mathbb{C}$ and $\mathbb{R}[\epsilon]$ them with different structure", which is an artificial choice. Maybe to illustrate the point : If I want to wor …

The following works constructively over an arbitrary local ring $R$ (constructively, $\mathbb{R}$ is a local ring). Assume that you matrix $M$ is canceled by a polynomial $Q$, of degree $m$ (with lea …

For any choice of $\lambda_1,\dots,\lambda_n$ there is an isomorphism: $$ k[X_1^{[\lambda_1]},\dots,X_n^{[\lambda_n]} ] \simeq k[Y_1^{[0]},\dots,Y_n^{[0]} ] $$ Which is given by $X_i \leftrightarro …

I'm looking for references for the following closely related facts: Given a Boolean algebra $B$, I denote by $\mathbb{Z}[B]$ the free ring generated by symbols $e_b$ such that $e_b e_{b'} = e_{b \cap …

Here is a method which is very efficient in the case were "constructive" is interpreted as "no axiom of choice at all, not even countable and no law of excluded middle", i.e. essentially "topos logic" …

I will show that this is not the case for $F=\mathbb{F}_9$. The proof generalize to any $\mathbb{F}_{p^k}$, with $k >1$. I'm starting from the observation that: $\mathbb{F}_9 \simeq \mathbb{Z}[i]/(3) …

Theorem: Given $A$ a boolean ring/boolean algebra then there is an equivalence of categories between the category of $A$-modules and the category of sheaves of $\mathbb{F}_2$-vector spaces on Spec $A$ …