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

learn more… | top users | synonyms (1)

1
vote
0answers
6 views

There always exists a finite, increasing chain of R-submodules of M isomorphic to $R/P$. Can we describe $P?

So I've been studying some commutative algebra and I came across the following theorem Theorem : Let R be a Noetherian ring. Let M be a non trivial R-module. There exists a chain of R-submodules ...
2
votes
1answer
19 views

Saturation of homogeneous ideal

Let $I \subset S=k[x_0,...,x_n]$ be a homogenous ideal. The saturation of $I$, $\bar{I}$ is defined to be $\{s \in S: \exists m \; s.t. \; \forall i \; x_i^m s \in I\}$ Is it true that $\bar{I}=\{s ...
1
vote
0answers
30 views

How do ring theorists think about square roots?

Let $R$ denote a commutative ring. Then it seems to me that we can to $R$ a square-root of $4$ as follows: $$R[\sqrt{4}] = R[x]/(x^2-4)$$ This defines a functor $\mathbf{CRing} \rightarrow ...
0
votes
2answers
40 views

Help in showing that the cusp $(y^2-x^3)\subset \mathbb{C}^2$ is not isomorphic to $\mathbb{C}$

Let $X:=(y^2-x^3)\subset \mathbb{C}^2$ be the vanishing of the polynomial $f(x,y)=y^2-x^3.$ I have proved an exercise in Hartshorne: If $\varphi:\mathbb{C} \to X, \ t \mapsto (t^2,t^3)$ is the ...
0
votes
0answers
16 views

Relationship of maximal ideals and associated primes in the total ring of fractions

Let $R$ be a ring, $S\subset R$ the set of non-zero-divisors. The total ring of fractions is then defined as $Q(R):=S^{-1}R$. Now I want to show that $A{ss}_R(R)$ being finite implies that there are ...
3
votes
0answers
31 views

Algebraically Closed Quotient Fields

It is well-known that if the quotient field of a commutative noetherian integrally closed domain $R$ is algebraically closed, then $R$ is a field. The proof is easy: let $r_0 \in R$ and choose $r_i ...
0
votes
0answers
29 views

Characterization of prime homogeneous ideals

Let $R$ be a graded ring and $I$ ideal in $R$ and homogeneous. $I$ is prime if and only if for all $a, b\in R$ homogeneous such that $ab\in I$ then $a\in I$ or $b\in I$. Let $ab\in I$ and $a ...
0
votes
1answer
18 views

Reference Request-Essential Extension

Let $R$ be a commutative ring with unit. Assume $R$ is an essential extension of each of its non-zero ideals. I feel that there should be something in the literature about this, but I could not find ...
3
votes
2answers
52 views

Showing the ideal $\left \langle yz,xz,yx+ay,x^2+ax \right \rangle$ is radical for all $a\neq 0$

Let $I_a = \left \langle yz,xz,yx+ay,x^2+ax \right \rangle$ be an ideal of $k[x,y,z]$, where $a \neq 0$. Show that $I_a$ is radical. What is the geometric meaning of the elements in ...
1
vote
0answers
38 views

What is extension of scalars used for in algebraic geometry?

Given a ring homomorphism $f:A \rightarrow B$ and an $A$-module $M$, one can construct and $A$-module with the tensor product: $M_B=B \otimes_A M$ which has a $B$-module structure. This is said to be ...
1
vote
1answer
33 views

$B/I$ and $B/J$ flat $A$-algebras; does $I=J$ hold?

Let $A\to B$ be a ring homomorphism. Consider $I$ and $J$ ideals of $B$ such that $B/I$ and $B/J$ are flat $A$-algebras. We know furthermore that there exists a non zero-divisor $t\in A$ such that ...
1
vote
2answers
35 views

Flat module and finite intersection of submodules

Let $R$ be an integral domain, $F$ be a flat $R$-module, and $A$ and $B$ are two $R$-submodules of $Q$, where $Q$ is the quotient field of $R$. How can we show that $F\otimes (A \cap B) = (F\otimes A) ...
1
vote
1answer
24 views

Singular ideals and rings

In Lam's book, Corollary (7.4)(2) says that for a nonzero ring $R$ we have $Z(R_R)≠ R$, where $Z(R_R) $ stands for the singular ideal of $R$.. But, some nonzero commutative rings are "singular" in the ...
1
vote
1answer
35 views

Zeros of specialization of a family of polynomials

Let $k$ be an algebraically closed field, and $K\supset k$ be an algebraically closed extension. Let $a\in K^n$ be a tuple, we call $a^\prime\in k^n$ a specialization of $a$ if for any $f(X)\in k[X]$ ...
1
vote
3answers
72 views

Recommended books on commutative algebra stressing links with algebraic geometry

Can someone recommend some books on commutative algebra stressing links with algebraic geometry? My concern is this. It seems to me that most of commutative algebra was formulated at least initially ...
1
vote
0answers
40 views

Is there an accepted definition of coprimality in commutative ring theory?

I can think of at least three possible definitions of coprimality in commutative ring theory: call $a,b \in R$ are coprime iff if $c \mid a$ and $c \mid b$, then $c \mid 1$. if $a \mid c$ and $b ...
0
votes
0answers
10 views

Integral basis of an extension of complete fields

Let $\mathcal{O}_K$ be a complete discrete valuation ring with quotient field $K = \text{Quot}(A)$. Let $L | K$ be an arbitrary finite field extension. Because $K$ is henselian, the integral closure ...
1
vote
1answer
91 views

normalization of a curve, simplest example

I am learning about normalization of nodal curves and I am trying to understand the simplest example: $xy=0$ As far as I understand its coordinate ring is $k[x]\oplus k[y]$ (let $k$ be an ...
3
votes
1answer
110 views

Problem about Gröbner basis.

I'd really appreciate if someone could help me. The problem is the following: Let $\psi_1,...,\psi_m \in k[x_1,\dots,x_n]$ and consider the $k$-algebra homomorphism: ...
1
vote
1answer
32 views

Characterization of Groebner Bases in terms of uniqueness of remainders

Let $I$ be an ideal of a polynomial ring $R=k[x_1,\ldots,x_n]$ over a field $k$. A Groebner basis of $I$ is a finite generating set $\{g_1,\ldots,g_m\}$ such that every leading monomial (according to ...
4
votes
2answers
168 views

Localisation and prime ideals

If $A$ is a ring and $S=\{1,f,f^2,f^3,...\}$ a multiplicative set of $A$, prove that $\mathrm{Spec}(A_f)=\mathfrak{V}((f))^c$. Notation: $A_f=S^{-1}A$ and $\mathfrak{V}((f))=\{P \in ...
5
votes
1answer
104 views

Primary descomposition of ideals

I'd appreciate if someone could help me a bit with this problem. Considering $\mathfrak{p}=(x,y), \mathfrak{q}=(x,z)$ and $\mathfrak{m}=(x,y,z)$ ideals in $k[x,y,z], k$ field. Is ...
0
votes
1answer
70 views

Why can one say WLOG assume $R$ is a local ring in Atiyah and MacDonald's 3.15 Exercise?

In Exercise 3.15 in Atiyah and Macdonald's Introduction to Commutative Algebra, the ring $R$ can be assumed to be a local ring, because of proposition 3.9. That proposition states that if $\phi: M ...
2
votes
2answers
66 views

Correspondence between nilpotents and between idempotents

It is well-known and easily proved that whenever $R$ is a commutative ring with unity and $S$ is a multiplicative subset of $R$, each ideal of the localization ring $R_S$ is an extended ideal (with ...
5
votes
2answers
103 views

Idempotent ideals in certain commutative rings

Let $R$ be a commutative ring with zero Jacobson radical such that each maximal ideal of $R$ is idempotent. Does it guarantee that each ideal is idempotent? I know only that if each maximal ideal ...
2
votes
1answer
309 views

Any ring is integral over the subring of invariants under a finite group action

I need to prove that if $G$ is a finite group that acts on ring $A$, and $A^G$ is the subring consisting of elements of $A$ which are invariant under all $g\in G$, then $A$ is integral over $A^G$. ...
1
vote
1answer
49 views

localized at associated prime of an ideal [duplicate]

The problem is as follows: Let $I\subseteq J$ be ideals in a Noetherian ring. Show that if $I_{p}=J_{p}$ for every associated prime $p$ of $I$,then $I=J$. It seems reasonable to consider ...
2
votes
0answers
46 views

Non-zero ideal in algebraic integers generated by two elements

I've been doing past questions for my exams next week and would like to check an answer: Let $I$ be a non-zero ideal of the algebraic integers and let $0\neq a \in I$. Show that $\exists b \in I$ ...
0
votes
0answers
42 views

Conjugation in algebraic number theory

Let $K$ be an algebraic number field of deg $n$ over $\mathbb Q$, then given $\alpha \in$ $O_k$ its ring of integers, we can choose a $\mathbb Q$-basis $\omega_1, \omega_2, ...,\omega_n$ of $K$ s.t. ...
0
votes
0answers
57 views

Traces of powers of a matrix $A$ over an algebra are zero implies $A$ nilpotent.

I would like to have a result similar to "Traces of all positive powers of a matrix are zero implies it is nilpotent". Namely: Let $R$ be a commutative $\mathbb{C}$-algebra, $A \in ...
4
votes
2answers
49 views

Integrally Closed domain and Principal Ideal

Let $R$ be an integrally closed local domain. Suppose there is a $y\in I^n$ such that $yI^n=I^{2n}$ for some $n$. I would like to prove that $I^n=(y)$. Source: The above question comes from the ...
3
votes
1answer
54 views

What does $(0:x)$ mean?

The following excerpt is from Eisenbud's "Commutative Algebra with a view toward Algebraic Geometry" on pg. 424 We can decide whether an element $x\in R$ is a nonzerodivisor from the homology of ...
1
vote
1answer
92 views

Can I use Krull dimension to test if a sequence of polynomials is regular?

A sequence $(f_1, \ldots, f_n)$ of elements of a commutative ring $R$ is said to be regular if for each $i$, $f_i$ is not a zero divisor in $R/(f_1, \ldots, f_{i-1})$. Call a sequence dimension ...
0
votes
2answers
211 views

A criterion of flat modules

Let $R$ be a commutative ring and $M$ an $R$-module such that for every ideal $I \subset R$ the natural map $I \otimes_R M \rightarrow IM$ is an isomorphism. Why is $M$ flat ? This result is ...
2
votes
1answer
48 views

Is every “prefield” a field?

Definition 0. Call a poset $P$ well-ranked iff it is well-founded, and for all $x \in P$, we have that any two maximal subchains in the lowerset generated by $x$ have the same length. ...
0
votes
0answers
28 views

Analytical isomorphism implies same multiplicities [duplicate]

I want to prove the following problem in Robin Hartshorne's Algebraic Geometry Chapter 1 exercise 5.14 If $P\in Y$ and $Q\in Z$ are analytically isomorphic plane curve singularities, show that the ...
-1
votes
1answer
22 views

Regular element of a Noetherian ring [duplicate]

Let $R$ be a Noetherian ring and $x\in R$ an $R-\mathrm{regular}$ element. Show that $\mathrm{Ass}_R(R/(x^n))=\mathrm{Ass}_R(R/(x))$ for every $n\geqslant 1$. Let $M$ be an $R-\mathrm{module}$. An ...
3
votes
2answers
120 views

Powers of prime ideals

I was reading through Atiyah-MacDonald and they mention that if a ring $A$ is a Noetherian domain of dimension 1 has the property that every primary ideal is equal to the product of a prime ideal ...
2
votes
0answers
29 views

What came first: pythagoras number or pythagorean fields? [migrated]

Which concept was first introduced: the pythagoras number of a field or pythagorean fields? I have not found anything on this matter, but my gut feeling says the latter. One can more directly link the ...
1
vote
1answer
44 views

Existence of homogeneous non-unit non-zero divisor in a particular graded ring.

Let $R$ be a finitely generated $k$-algebra of dimension greater than $1$, let $Q$ be any maximal ideal of $R$. It is claimed by my lecturer that one can find a homogeneous, non-unit, non-zero divisor ...
0
votes
1answer
75 views

In $A$-Mod, $M\oplus A\cong A\oplus A$ implies $M\cong A$

(Exercise from an introductory course in homological algebra) Whenever $A$ is a commutative ring with unit and $M$ an $A$-module, the following holds: $$M\oplus A\cong A\oplus A \Rightarrow ...
1
vote
1answer
50 views

Easy explanation on primary decomposition of ideals. [duplicate]

The primary decomposition of an ideal $(x^2, xy)$ is $$(x^2, xy) = (x) \cap (x, y)^2$$ which can be found on these notes. Could someone explain to me how this can be done? Edited: My question ...
10
votes
1answer
741 views

What conditions guarantee that all maximal ideals have the same height?

It fails in general that all maximal ideals in a commutative ring with unity have the same height. It's easy to construct a counter-example when the ring is NOT an integral domain (consider the ...
1
vote
1answer
28 views

Exact sequence of graded modules and localization

I know that a sequence of modules is exact iff the localization at each prime ideal is exact What happens in the case we are working with graded modules? Can we say that a sequence is exact iff the ...
0
votes
1answer
38 views

Flatness of quotient rings

The following is Exercise 2.4, in Chapter 1 of Liu, Algebraic Geometry and Arithmetic Curves: Let $I$ be a finitely generated ideal of $A$: $A/I$ is flat. $I^2 = I$. $I = (e)$ where $e^2=e$. I ...
8
votes
1answer
181 views

Is $R/N(R)$ a faithfully flat $R$-module?

I'm studying recently faithfully flat modules and I'd like to know the following: Is $R/N$ faithfully flat as $R$-module, where $R$ is a commutative ring with unit and $N$ is the ideal of ...
6
votes
2answers
316 views

$M\oplus A \cong A\oplus A$ implies $M\cong A$?

Let $A$ be a commutative unital ring and $M$ an $A$-module. Suppose that $M\oplus A \cong A\oplus A$. Then is $M\cong A$? We have that both $M\oplus A$ and $A\oplus A$ are biproduct for $(A, A)$ ...
1
vote
0answers
30 views

Local ring of an affine curve $K$ at a point $p\in K$

I'm reading A Royal Road to Algebraic Geometry by Holme. The book defines the local ring as follows: The local ring of $K$ at $P=(a,b)$ is the ring ...
2
votes
1answer
29 views

Hilbert function and Hilbert polynomial

I have largely studied Hilbert function and Hilbert polynomial for polynomial rings over fields of characteristic zero. Is it possible to extend the theory also for polynomial rings over fields of ...
5
votes
1answer
192 views

Primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field

I am looking for the primary decomposition of $(x^2,xy,xz)$ in $k[x,y,z]$ where $k$ is a field. I am not looking for a solution here, rather a hint or two. Is there a general strategy for ...