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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.

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$^fR$ is a self-dual bimodule in Gorenstein local rings

Let $(R,m)$ be a Gorenstein local ring of characteristic $p$. Let $^f R$ denote the $R-R$-bimodule with additve group $R$ and left and right scalar multiplication given by $a\circ r\bullet b = a^p rb$. Then $\text{Hom}_R(^fR,R)\cong\ ^fR$. Since…
Bromelain
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Length of $\text{Tor}$ modules in complete intersection rings with characteristic $p$.

Let $(R,m)$ be a complete intersection ring with characteristic $p$. If $M$ is a finitely generated $R$-module with finite length i.e. $l(M)<\infty$. I know that $l(F(M))\geq p^nl(M)$, where $n=\dim R$. $(\text{The functor $F$ is the Frobenius…
Display name
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$A$ local ring and $M,N$ be finitely generated $A$-modules $\phi: M \to N$ be homomorphism

I am given that $A$ a local ring, $\mathfrak{m}$ its maximal ideal and $M,N$ be finitely generated $A$-modules. $\phi: M \to N$ be homomorphism. I have to show that $\tilde{\phi}: M/\mathfrak{m}M\to N/\mathfrak{m}N$ induces a homomorphism of…
Mike192000
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Nilradical and showing that it is a local ring of $\mathbb{Q}[x,y]/(x^2, xy, y^3)$

Is it true that the nilradical is generated by $(x)$ and $(y)$? My intuition says yes because $(ax+by)^2=a^2x^2+2abxy+y^2=y^2$ and thus $(ax+by)^3=axy^2+by^3=0$ so every combination of $x,y$ will be nilpotent but am I forgetting other elements? Also…
Mike192000
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Question about localization by maximal ideals

I have some questions about localization by maximal ideals. Let $A$ be commutative ring and $p ⊂ A$ be a prime ideal. then $A_p$ is local ring with maximal ideal $m = pA_p$. Then I have question that (i) $m^2 = p^2A_p$ is it correct? Nextly $R$ be…
Shintaro
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Coalgebra structure on the UEA of a Lie Rinehart algebra

Good evening, $R$ is a commutative ring and $A$ is a commutative $R$-algebra. Let $(L,\rho)$ a $(R,A)$-Lie Rinehart algebra. $L$ is a $R$-Lie algebra with a $A$-module structure and $\rho:L\longrightarrow Der_R(A)$ is a morphism of $R$-Lie algebra…
Damdom
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Product of elements of Dedekind domain contained in powers of a prime ideal

If $a,b\in R$ are nonzero in Dedekind domain $R$, and $\mathfrak{p}$ a prime ideal. Suppose $m=sup\{k:a\in \mathfrak{p}^k\}$, $n=sup\{k:b\in \mathfrak{p}^k\}$. I want to show $m+n=sup \{k:ab\in \mathfrak{p}^k\}$. I know that both $m,n$ are finite,…
Jun Xu
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Are Fitting ideals of a finite flat module flat?

Let $A$ be a commutative unital ring. Let $M$ be a finite flat $A$-module. We can present $M$ as the cokernel $$A^{\oplus J}\xrightarrow{\varphi} A^n\to M\to0$$ and there exists $a_{ij}$ for $1\leq i\leq n$ and $j\in J$ such that…
Display Name
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Atiyah-Macdonald Lemma 5.19

Let $K$ be a field, $\Omega$ an algebraically closed field. Let $\Sigma$ be the set of all pairs $(A,f)$ where $A$ is a subring of $K$ and $F$ is a homomorphism of $A$ into $\Omega$. We partially order $\Sigma$ as follows: $$(A,f) \leq (A',f') \iff…
shintuku
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ideals and ring extensions

Let $R_1$ be the ring of polynomials (over $\mathbb{C}$) and $R_2$ a finite free module over $R_1$ (generated by elements $b_1,\ldots,b_m$). Now I consider an ideal $I\subset R_2$ and the ideal $J=I\cap R_1$ and consider the corresponding quotients…
user382144
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Localization of direct sum of rings

Let $R$ be a ring, $m$ is a maximal ideal of $R$. Thereofre, $R\oplus m$ is a maximal ideal of $R\oplus R$. What is the local ring $(R\oplus R)_{(R\oplus m)}$? I think this involves a component of $R_R$ which makes no sense to me.
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Morphism from a Jacobson ring to an arbitrary ring; preimage of a maximal ideal is maximal?

Recall that a Jacobson ring $A$ is a ring such that all prime ideals are an intersection of maximal ideals. Is the following true? If $f:A\to R$ is a ring map with $A$ Jacobson, and $\mathfrak m$ is a maximal ideal of $R$, then $f^{-1}\mathfrak m$…
Nico
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finite locally free modules over a polynomial ring $R[t]$

Let $f:M\to N$ be a homomorphism of finite locally free $R[t]$-modules, where $R$ is a ring, and assume $P = \operatorname{coker} f$ is a f.g. $R$-module. I want to show that $P$ is finitely presented over $R$. Since $t$ is an endomorphism of the…
ALMOST_COMPLEX
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Kernel of a specific map

This question is a generalisation of this question. Suppose that $k$ is a commutative ring (we could assume more that it is a field), and suppose that $R$ is a commutative $k$-algebra. Given any $\sigma\in\mathfrak{S}_n$, we could view it as a map…
Guanyu Li
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What does it mean for $B$ to be a local ring of $A$?

In Atiyah and Macdonald, an exercise states: Let A be a valuation ring of a field K. Show that every subring of K which contains A is a local ring of A. I tried looking up what this means and found two (a priori) different characterizations. One…
mattematician
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