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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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Question on Comaximal Ideals

There's a small detail in a proof of the Chinese Remainder Theorem for modules I don't understand when it comes to showing the normally constructed homomorphism is a surjection. Suppose $R$ is a commutative unital ring, with $I_1,\dots,I_k$…
Bongle
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Torsion-free quotient of integer polynomial ring

Consider the ring of polynomials $\mathbb{Z}[x,y]$ and let $I$ be the ideal $(xy,x+y)$. Is the quotient $\mathbb{Z}[x,y]/I$ torsion-free as a $\mathbb{Z}$-module? How does one approach this type of question in the more general case when $I$ is…
Bok Mal
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Nonintegral element and a homomorphism

Assume $R\subseteq S$ are rings. Choose $x\in S$ nonintegral over $R$. I want to define a homomorphism from $R[x^{-1}]$ to a field which maps $x^{-1}$ to zero. I was trying to show that $R[x^{-1}]$ is a polynomial ring in one variable. Then I…
H.W.
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Length of regular local ring

Let $A$ be a regular local ring of dimension $n$ with maximal ideal $\mathcal m$. Then one can consider the $A-$module $A/\mathcal m^i$ for all natural numbers $i$. My question is simply what length this module has. For $i=0$ it is clearly of length…
Cyril
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Can the completion of a non-domain be a domain

Suppose, $(R,m)$ is a Noetherian ring that is not a domain. Can $\hat{R}_m$ be a domain? I think this cannot be the case for if $a,b\in R$ s.t. $a,b\neq 0$ and $ab=0$. Then, this $R$ is a Noetherian local ring, by the Krull intersection theorem,…
Raul D
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Two points in a proof of regularity of $R/I$

In the proof of the fact that "if $I$ is an ideal of the regular local ring $(R,m)$ such that $R/I$ is regular then $I$ can be generated by part of a minimal generating set of of $m$", I saw in a textbook that the author takes the dimension of…
karparvar
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Characterization of Discrete Valuation Rings

Let $R$ be a Noetherian local domain with unique maximal ideal $M$. Then I want to show that if every $M$-primary ideal is a power of $M$, then $R$ is a Discrete Valuation Ring. I know I'll be done if I can show that $M$ is principal, or that $M$…
Nishant
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About weakly associated primes

Let $A$ be a commutative ring, and $M$ an $A$-module. A prime ideal $\mathfrak{p}\subset A$ is said to be weakly associated to $M$ if it is minimal over some $\operatorname{ann}m$, where $m\in M$. I came across a statement that…
ashpool
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Product of a complete module and a finite module

Let $A$ be a commutative noetherian ring, $I$-adically complete (and separated) with respect to an ideal $I \subseteq A$. Let $M$ be a finite $A$-module, and let $N$ be an $I$-adically complete $A$-module. Is it true that $M\otimes_A N$ is also…
the L
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Projective syzygy vs. free syzygy

When referring to syzygies, some books refer to free resolution and some books refer to projective resolution. Are they equivalent in some sense? Is it true, for instance, that the $n$-th syzygy in a finite projective resolution is the $n$-th…
ashpool
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intersection of non zero prime ideals of polynomial ring R[x] over integral domain R is zero

Let R be an integral domain. Then how to show that intersection of non zero prime ideals of R[x] is zero.
user156736
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Depth of finite projective modules over a nonlocal ring

Let $(A,\mathfrak{m})$ is a Noetherian local ring and $P\neq0$ is a finitely generated projective $A$-module. Then by Auslander-Buchbaum formula, $\operatorname{depth}P=\operatorname{depth} A$. But is this true even if the ring is not local? That…
ashpool
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Can the submodule generated by action of nilradical be equal to whole module

Let $A$ be a commutative ring with unity, $N$ be the nilradical of $A$, $M$ be an $A$-module. Is it always true that $NM$ is a proper submodule of $M$? If $M$ is finitely generated then by Nakayamma Lemma $NM$ must be a proper submodule. Are there…
bharath
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Is the integral closure of local domain a local ring?

Suppose $A$ is a local domain, with field of fractions $K$, let $A'$ be the integral closure of $A$ in $K$, is $A'$ a local ring?
user93417
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Question on finitely presented algebra

Suppose $S$ is a finitely presented $R$-algebra. If $g:R[x_1, \ldots, x_n] \to S$ is surjective, then $\ker(g)$ is finitely generated. We can write $S$ as $R[y_1, \ldots, y_m]/(f_1,\ldots,f_t)$ and write $g$ as $g:R[x_1,\ldots,x_n] \to R[y_1,…
afzd
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