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2
votes
0answers
62 views

non-Noetherian r-Noetherian ring with Noetherian total quotient ring

A commutative ring is said to be r-Noetherian if every regular ideal is finitely generated, where an ideal is said to be regular if it contains a non-zerodivisor. Does there exist a non-Noetherian ...
53
votes
1answer
1k views

$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$

Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$? This is a ring-theoretic analog of my previous question about abelian groups: In fact, in any algebraic category we may ...
3
votes
0answers
100 views

rings with 'flat functions'

Let $(R,\mathfrak{m})$ be a local ring over a field. Suppose the ring has flat elements, i.e. $\mathfrak{m}^\infty\neq\{0\}$. (The prototype is of course $C^\infty(\Bbb{R}^p,0)$, or a quotient of it, ...
8
votes
2answers
383 views

Strategies for proving a category is Noetherian?

Let $C$ be a small linear category over a commutative ring $R$. A representation of $C$ is an $R$-linear functor $C \to \mathrm{Mod}(R)$. For example, for each $c\in C$, there is a representation ...
0
votes
1answer
134 views

$\inf\{i\in \mathbb N \cup \{0\}\cup\infty\mid Ext^i_R(R/I,R)\neq 0\}=0 ?$

Let $R := k[x_1, \cdots, x_n, \cdots]/(x_1^1, \cdots , x^n_n, \cdots),$ where $k$ is a field. Set $I:=(x_1, \cdots, x_n, \cdots)$. the questions are: Is $\inf\{i\in \mathbb N \cup \{0\}\cup ...
2
votes
0answers
186 views

Weak assassins and essential morphisms

Let $R$ be a commutative ring and let $M\rightarrow N$ be an essential morphism of $R$-modules. Then, $M$ and $N$ have the same associated primes. Over non-noetherian rings the notion of associated ...
3
votes
2answers
462 views

Condition for a local ring whose maximal ideal is principal to be Noetherian

The statement "a local ring whose maximal ideal is principal is Noetherian" is (I think) false. The ring of germs about $0$ of $C^\infty$ functions on the real line seems to be a counterexample since ...
2
votes
1answer
304 views

quotients of varieties as non-noetherian schemes?

Let $X$ be a variety (i.e. a reduced scheme of finite type over a field) and let $G$ be an abstract group, finitely generated, acting of $X$ algebraically freely. The example I have in mind is ...
2
votes
1answer
195 views

Is the pushforward via a proper map of a finite presentation module of finite presentation?

It's true that the pushforward of a coherent sheaf is coherent via a proper morphism: but do proper morphisms preserve a finite presentation? Under some assumptions perhaps? Does it change if we are ...