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Commutative rings, modules, ideals, homological algebra, computational aspects, invariant theory, connections to algebraic geometry and combinatorics.

1 vote

Restricted extension of prime ideals of the ring of polynomials over $\mathbb{Q}$

I think the answer is yes when $n=2$. Let $R=\mathbb{Q}[x,y]$; since $R$ is a two-dimensional the only non-trivial case is for height-1 primes, and since $R$ is a UFD we can suppose that the prime id …
Dario Spirito's user avatar
-1 votes

UFD containing element with finite quotient

Edit: disregard this answer, I completely misread the question. Let $D:=\mathbb{Z}[X]$, and let $S:=D\setminus(2D\cup(3,X)D)$. Let Take a DVR $V$, and let $R=V[X]$ be the polynomial ring on $V$. Th …
Dario Spirito's user avatar
5 votes
Accepted

Are unique prime ideal factorization domains locally noetherian?

No, this property does not imply that $A$ is locally Noetherian. For example, let $F\subset L$ be an extension of fields, and let $A=F+XL[[X]]$ (that is, $A$ is the set of power series over $L$ whose …
Dario Spirito's user avatar
1 vote
Accepted

Chain of closed irreducible sets on Zariski Riemann spaces

The answer to the problem is no. One reason is that the dimension of a valuation ring $V\in\mathrm{ZR}(K,A)$ may be greater than the dimension of $A$. (The supremum of the dimension of the elements of …
Dario Spirito's user avatar