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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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Equivalence of elements in a ring of fractions

Let $R$ be a commutative ring and $S \subseteq R$ its multiplicative subset. The equivalence relation on $R \times S$ used in the definition of the ring of fractions $RS^{-1}$ is defined as follows: $(r,s) \sim (r',s')$ iff there exists $x \in S$…
pepa.dvorak
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Jacobson radical of rings

I have some question about the Jacobson radical of rings. What is $J(R)$ when $R$ is a Principal Ideal Domain but not a field? e.g. I know that $\mathbb Z$ is a PID and why is $J(\mathbb Z)=0$ but can we say that is true for every Principal Ideal…
user130225
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Appropriate notion of localization of a Galois ring extension

Earlier, I had a asked a question for a notion of Galois ring extension. I was particularly interested in Peter Patzt's answer. So, given an integral domain $R$ with field of fractions $F$ and a Galois extension $L$ of $K$, let $T$ be the integral…
B M
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When do we have $\operatorname{depth}_{A} B = \operatorname{depth}_B B$?

Let $(A,\mathfrak{m}) \to (B,\mathfrak{n})$ be a local homomorphism with $A$ a regular local ring. Assume further that this ring map is finite. How can we prove that $\operatorname{depth}B = \operatorname{depth}_A B$? What about if we replace $B$…
user38268
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a submodule of $R^n$

Let $R$ be a commutative ring of positive characteristic p. If $M$ is a submodule of $R^n$, let $M^{[p]}$ be the submodule of $R^n$ generated by $(a_1^p,\cdots,a_n^p)$ where $(a_1,\cdots,a_n)\in M$. The question is: Is $M^{[p]}$ independent of the…
Yubin
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where is the mistake in this "paradox"?

In the middle of page 34 in Bruns and Herzog, Cohen-Macaulay Rings, the authors present the following situation: Let $k$ be a field and let $R=k[X,Y]$ be a graded ring with grading induced by $\operatorname{deg}X=0, \operatorname{deg} Y=1$. Then…
Manos
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Prove that $A[x]$ is a flat $A$-algebra.

This is from exercice 5, chap 2 from Atiyah and McDonald "Introduction to Commutative Algebra". Let $A[x]$ be the ring of polynomials in one indeterminate over a ring $A$. Prove that $A[x]$ is a flat $A$-algebra. Clearly, we notice that…
ALM
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Looking for a more elegant / generic proof of the reducibility of a polynomial in $K[[X,Y]]$

The polynomial $P(X,Y)=XY-(X+Y)(X^2+Y^2)$ is irreducible in $K[X,Y]$, as a sum of two homogenous forms of degree 2 and 3 ($K$ is supposed to be algebraically closed). To look at the irreducibility in $K[[X,Y]]$, I first worked in $K[[X]][Z]$, where…
brunoh
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Example showing why Macaulay's lemma doesn't work for inhomogeneous ideals

Macaulay's lemma states: Let R be a polynomial ring and I a homogeneous ideal. Then the Hilbert function of I is the same as the Hilbert function of in(I). (Schenck, Computational Algebraic Geometry, p55) (Where in(I) / lt(I) is the ideal consisting…
DavidA
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What is the usefulness of $M[X]$?

Given a commutative ring with unity $R$ and a $R$-module $M$, it can be defined the $R[X]$-module (here $X$ is an indeterminate) $M[X]$ as the set the formal expressions $\sum_{k=0}^nm_kX^k$, with $m_k\in M$, endowed with the natural sum and scalar…
Matemáticos Chibchas
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Finding the Hilbert Function for a certain ring

Right now I'm trying to find the Hilbert Function , and the corresponding Hilbert Polynomial for the ring $M=k[x,y,z,w]/(x,y) \cap (z,w)$. I just finished reading the first chapter of Eisenbud, so I don't have that much of an advanced toolbox for…
Dedalus
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An invariant of homogeneous ideals in polynomial rings

Let $k$ be a field, $S=k[x_1,\dots,x_n]$ and $I$ a homogeneous ideal of $S$. Let $f_1,\dots,f_l$ be a minimal generating set of $I$ and let $d$ be the maximal degree among the degrees of the $f_i$. Then $d$ is an invariant of $I$. Eisenbud in…
Manos
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The number of prime ideals lying over a given prime ideal

Put $A = k[x]$, where $k$ is an algebraically closed field and $x$ is an indeterminate. Let $B$ be a ring and $f: A \rightarrow B$ be finite integral morphism. How can one show that the number of prime ideals of $B$ which lie over a given prime…
Chan Yat-Fay
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Regularity of a basis over a local ring

The following Theorem (Thm. 19.9 in Matsumura, CRT) will be necessary for the understanding of my question: Theorem: Let $A$ be a Noetherian local ring and $I$ a proper ideal of $A$ with $\operatorname{projdim} I < \infty$. Then $I$ is generated by…
Manos
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Domain where for each ideal $I$ and each $a \in I$, $I = \langle a,b \rangle$ is Dedekind?

Say $R$ is a domain such that for any nonzero ideal $I$ of $R$ and any $a \in I$, we have $I = \langle a,b\rangle$ for some $b \in I$. Is $R$ a Dedekind domain? If not, what additional assumptions do I need to force it to be one?
Sam Hopkins
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