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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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Examples demonstrating that the finitely generated hypothesis in Nakayama's lemma is necessary

Recall that Nakayama's lemma states that Let $R$ be a commutative ring with unity, and let $J$ be the Jacobson radical of $R$ (the intersection of all the maximal ideals of $R$). For any finitely generated $R$-module $M$, if $IM=M$ for some ideal…
Zev Chonoles
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Exercise 2.17(d) of Eisenbud's Commutative Algebra

First some notation: Let $P$ be a homogeneous prime ideal of a $\Bbb{Z}$ - graded ring $R$, $U$ the multiplicative subset of all homogeneous elements not in $P$. Suppose that there exists a homogeneous element $f$ of degree $1$ that is not in…
user38268
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Spectrum of a ring is irreducible if and only if nilradical is prime (Atiyah-Macdonald, Exercise 1.19)

Can anyone help me with this exercise, please? A topological space $X$ is said to be irreducible if $X\neq\emptyset$ and if every pair of non-empty open sets in $X$ intersect, or equivalently, if every non-empty open set is dense in $X$. Show that…
user103652
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Nilradical of polynomial ring

Let $R$ be a commutative ring. The nilradical $\text{nil}(R)$ is the set of all nilpotent elements, and it is the intersection of all the prime ideals of $R$. Is the following true in the polynomial…
PatrickR
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Does inclusion of a ring into a polynomial ring induce a closed map on prime spectra?

Let $A$ be a commutative (unital) ring, and $A[x_1,\ldots,x_n]$ a polynomial ring over it in some finite number of variables. The inclusion $i\colon A \hookrightarrow A[x_1,\ldots,x_n]$ induces (by contraction) a continuous surjection…
jdc
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Exercise 11.5 from Atiyah-MacDonald: Hilbert-Serre theorem and Grothendieck group

I don't understand Exercise 11.5 of Atiyah & MacDonald, which demands one elaborate upon or rephrase the Hilbert–Serre Theorem (11.1) in terms of the Grothendieck group $K(A_0)$. Here's the set-up in more detail. $A$ is a commutative Noetherian…
jdc
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Why does the structure theorem for finitely generated modules over PIDs fail for arbitrary modules over a PID?

The proof that I know of the theorem goes like this: Any module $M$ is a quotient of a free module $F$ (over any ring). Any submodule $K$ of a free module $F$ over a PID $R$ is a free module, so in particular the kernel of the above quotient map is…
Vladimir Sotirov
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Integral domain with fraction field equal to $\mathbb{R}$

I wonder if there is an integral domain $A\subseteq \mathbb{R}$ which is not a field, and such that the field of fractions of $A$ is equal to $\mathbb{R}$? Edit: here as a possible direction: it is known that there is no finite field extension…
the L
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Do the zero divisors form an ideal?

The problem is to prove that, in a commutative ring with identity, the set of ideals in which every element is a zero divisor has a maximal element with respect the order of inclusion, and that every maximal element is prime. But I´m thinking* that…
Susuk
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Every module over a field is free. Is every commutative ring whose modules are all free a field?

Let $A$ denote a commutative ring. Then if $A$ is a field, we may deduce that every $A$-module is free. Does the converse hold? i.e. If every $A$-module is free, can we deduce that $A$ is a field?
goblin GONE
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Exactness of a short sequence of quotient modules

Suppose R is a commutative ring with 1, I $\subset R$ is an ideal. We have R-Modules A, B and C with C being flat, as well as a short exact sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ Consider the induced sequence $0…
Cedric B
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Kernel of map between polynomial rings that takes monomials to monomials

Let $k$ be a field (say of characteristic $0$). Let $z_1,\ldots,z_n \in k[y_1,\ldots,y_m]$ be monomials, and consider the ring homomorphism $\phi : k[x_1,\ldots,x_n] \rightarrow k[y_1,\ldots,y_m]$ defined by $\phi(x_i)=z_i$. There are many…
Adam Smith
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When some polynomials in $\mathbb Z[X]$ determine a regular sequence in $\mathbb Z[X_1,\dots,X_n]$?

Let $f_1,\dots,f_n\in\mathbb Z[X]$ be non-constant polynomials (not necessarily distinct). Is it true that $f_1(X_1),\dots,f_n(X_n)$ is a regular sequence in $\mathbb Z[X_1,\dots,X_n]$? The trivial case (that suggested me this question) is…
user26857
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Ideal correspondence

I'm confusing the ideal correspondence theorem. Is the following right? Ideal correspondence: Let $f:A \to B$ be a ring homomorphism. Then there is a one-to-one order-preserving correspondence between ideals of $f(A)$ and ideals of $A$ which…
Gobi
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Integral closure $\tilde{A}$ is flat over $A$, then $A$ is integrally closed

Question. Let $A$ be an integral domain and $\tilde{A}$ be its integral closure in the field of fractions $K$. Assume that $\tilde{A}$ is a finitely generated $A$-module. I want to prove that if $\tilde{A}$ is flat over $A$, then $A$ is integrally…
Karatuğ Ozan Bircan
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