Questions tagged [coherent-rings]

In mathematics, a (left) coherent ring is a ring in which every finitely generated left ideal is finitely presented.

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$I$ finitely presented nilpotent ideal of a commutative ring $R$ and $R/I$ coherent implies $R$ coherent

Let $R$ be a commutative ring, and let $I$ be an ideal. We assume that $I$ is nilpotent, so $I^n=0$ for some $n$. Moreover, we assume that it is finitely presented, namely it is the cokernel of some ...
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$M$ is a coherent left $R$-module implies $M/IM$ is a coherent $R/I$-module.

Do you know how to prove that for a ring $R$, and a bilateral ideal $I$ of $R$, if $M$ is a coherent left $R$-module, then $M/IM$ is a coherent $R/I$-module? I know how to prove the fact if $I$ ...
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1answer
82 views

Can we control the number of homogeneous generators of a f.g. homogeneous ideal?

Let $G$ be an abelian group and $R$ be a $G$-graded ring. Question $1$: Is there a map $\phi:\mathbb{N}\rightarrow\mathbb{N}$ such that for every $n\in \mathbb{N}$ and any homogeneous ideal $I$ ...
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1answer
53 views

Do two similar boolean matrices have same number of non-zero entries?

I was just wondering: is it necessarily the case that if $A$ is a $(0, 1)$ matrix, and $SAS^{-1}=B$, where $B$ is also a $(0,1)$ matrix, then do $A$ and $B$ have the same number of $1$s? I have the ...
2
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43 views

infinite indeterminates over a left noetherian ring is left coherent(want a basic proof)

A ring is left coherent if every finitely generated left ideal is finitely presented. Statement: If $R$ is a left noetherian ring, then $R[X]$ is left coherent where $X$ represents infinite many ...
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77 views

Coherent Ring, whose nilradical is not finitely generated.

Let $A$ be a commutative ring with $1$.We say that $A$ is coherent if and only if every finitely generated ideal of $A$ is finitely presented. Does there exist a coherent ring such that nil-radical ...
2
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1answer
730 views

Polynomial ring in infinitely many variables over a noetherian ring is coherent

If $R$ is noetherian, show that the polynomial ring of infinite variables $R[x_1,x_2,...]$ is coherent, i.e. every finitely generated ideal is finitely presented. I don't really know how to get ...
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92 views

(quasi)coherent rings for which $\dim R[T]\neq \dim R+1$

What are some some examples of (quasi)coherent rings for which $\dim R[T]\neq \dim R+1$? Why (hopefully geometrically) should we not always have equality? Notation. Let $I,J$ be two ideals of ...
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1answer
484 views

Is every ring/module the filtered colimit of its finitely presented/coherent/quasicoherent subrings/submodules?

Is every ring/module the filtered colimit of its coherent/quasicoherent subrings/submodules? What about finitely presented subobjects? What's the intuition behind each case? Notation. Let $I,J$ be ...
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240 views

Geometric intuition for coherent rings, modules, and sheaves

Throughout, all rings are commutative. Definition 1. A ring $R$ is coherent if the solutions $\mathbf x=(x_1,\dots,x_n)$ to a linear equation $\mathbf{rx}=0$ are a finitely generated $R$-submodule of ...