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Questions tagged [commutative-algebra]

Questions about commutative rings, their ideals, and their modules.

Commutative algebra is the area of mathematics that deals with commutative rings and their ideals, as well as modules over commutative rings.

Many results and tools of commutative algebra are cornerstones of algebraic geometry. Important tools of commutative algebra include localization and completion of rings and modules.

16857 questions
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What is the integral closure of the ring $k[t^3-t, t^2-1]$ in $k(t)$?

What is the integral closure of the ring $k[t^3-t, t^2-1]$ in $k(t)$, where $k(t)$ is the field of fractions for $k[t]$? I have read a post on this site saying that the integral closure is just $k[t]$, but I don't quite understand why. Thus if…
some1fromhell
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When does the ideal product commute with intersections?

Let $R$ be a commutative ring and let $\mathfrak{m} \subseteq R$ be a finitely generated ideal. If $(M_n)_{n \in \mathbb{N}}$ is a family of submodules of some $R$-module with $M_0 \supseteq M_1 \supseteq \dotsc$, do we have $\mathfrak{m} \cdot…
Martin Brandenburg
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Topology induced by a stable filtration on a ring

Let $R$ be a ring with a descending filtration by ideals $$R = I^{(0)} \supset I^{(1)} \supset I^{(2)} \supset \dots$$ such that $I^{(j)} I^{(k)} \subset I^{(j+k)}$, with equality whenever $k$ is sufficiently large. Is the topology induced by this…
isekaijin
  • 1,765
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Big subalgebras of the free polynomial algebra

Let $R$ be a Noetherian commutative unital ring. Let $n\geq 1$ be an integer. Suppose that $R[x_1, \dots, x_n]$ has a $R$-subalgebra $A$ such that $R[x_1, \dots, x_n]$ is a free finitely generated $A$-module. Is it true that $A$ is isomorphic to…
user693936
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A strong converse to Hilbert's basis theorem

Let $R$ be a commutative ring with a multiplicative identity such that there is a finitely generated $R$-algebra that is Noetherian. Is $R$ Noetherian then? I tried to prove this using the fact that the homomorphic image of a Noetherian ring is…
user693936
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A finitely generated algebra with the same number of prime ideals as the base ring is finitely presented

Let $f:A\rightarrow B$ be a map of integral domains making $B$ into a finitely generated $A$-algebra. Suppose that $f^{-1}$ induces a bijection from the set of prime ideals of $B$ to the set of prime ideals of $A$. Is $B$ a finitely presented…
user693936
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Automorphisms of discretely valued fields

Let $R$ be a DVR. Let $\phi$ be an automorphism of its fraction field. Can $R$ be a proper subset of $\phi(R)$? There definitely do exist DVRs that contain a unital subring isomorphic to themselves (e.g. $k[[x^2]]\subset k[[x]]$) but I am not sure…
user691994
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Quadratic extension of $\mathbb{Q}(X)$ generated by the square root of a square-free polynomial

Let $f(X)\in\mathbb{Q}[X]$ be a square-free polynomial, that is, not divisible by any prime of $\mathbb{Q}[X]$. Let $K:=\mathbb{Q}(X)[Y]$, where $Y$ is a square root of $f(X)$, or equivalently $Y$ solves the monic polynomial equation given by…
Federica Maggioni
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Atiyah Macdonald exercise 1.5

I am trying to prove that: The contraction of a maximal ideal $\mathfrak{m}$ of $A[[x]]]$ is a maximal ideal of $A$, and $\mathfrak{m}$ is generated by $\mathfrak{m}^c$ and $x$. I have made the following progress: Suppose $x\not\in \mathfrak{m}$.…
erdoswiles
  • 629
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Element of $T(R)$ that projects to element of $R/P$ for every minimal prime $P$ of $R$?

Let $R$ be a reduced ring and $T(R)$ the total ring of fractions of $R$ (i.e. localizing $R$ at nonzerodivisors). Any element of $T(R)$ maps naturally to an element of $T(R/P)$ since $a/b \in T(R)$ projects to $a + P / b+ P$ and $b \notin P$ since…
the.myth.of.math.and.meth
  • 150
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Representation of an element of the field of fractions of a Dedekind domain as a fraction of elements which are relatively prime to a given ideal

This is a generalization of this question. Let $A$ be a Dedekind domain. Let $K$ be the field of fractions of $A$. Let $I$ be a non-zero ideal of $A$. Let $\alpha$ be a non-zero element of $K$ which is relatively prime to $I$. That is, $(\alpha)$…
Makoto Kato
  • 42,602
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The principal fractional ideals of an integral domain form a directed partially ordered group

Let $R$ be an integral domain and $K$ be its quotient field. Let $G = \{aR: a\in K^{\times}\}$. Then $G$ is a partially ordered group under $aR\leq bR$ iff $bR\subseteq aR$. But I have hard time to show $G$ is a directed partially ordered group.
Sharma Sharma
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Residue fields of Gorenstein local rings have finite injective dimension?

If $(R,\mathfrak m,k)$ is a Gorenstein local ring, then show that $\textrm{inj dim}_R\ k$ is finite. This was previously asked here as a second part of the question and remained unaswered, but I think it is independent of the first part and…
user26857
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practical condition for minimality in primary decomposition

Situation: $I$ is an ideal in a polynomial ring with a primary decomposition, not necessarily minimal (minimal=irredundant). I want to minimal-ize it. For any couple of primary ideals with the same radical, I take the intersection of the two, and…
NeuWolferl
  • 83
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3 answers

Localization of Noetherian and Artinian Modules

Theorem: Let $R$ be a commutative ring with unity, and $S\subset R$ be a multiplicatively closed subset. If $M$ is a Noetherian (Artinian) $R$-module then $S^{-1}M$ is Noetherian (Artinian) $S^{-1}R$-module. I know the proof of the theorem, but my…
i.a.m
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