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

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-1
votes
1answer
13 views

Prove that T is not a zerodivisor in A[T]

Let A be any ring, consider the polynomial ring A[T]. Prove that T is not a zerodivisor in A[T]. Generalise the argument to prove that a monic polynomial $$ f=T^n+a_{n-1}T^{n-1}+\dots+a_0 $$ is ...
3
votes
1answer
145 views
+50

Zero divisors in $A[x_1,x_2,\dots,x_r]$

I am trying to show that if $f(x_1,x_2,\ldots, x_r) \in A[x_1,x_2,\ldots, x_r]$ is a zero divisor then there exists $a$ in $A-\{0\}$ such that $af=0$ in $A[x_1,x_2,\ldots, x_r]$. What I have ...
2
votes
2answers
51 views

Going-up and going-down theorems: motivation

I am reading about the going-up and going-down theorems in Atiyah & Macdonald's commutative algebra book. I'm wondering if anyone could give me some basic facts/examples to help me understand why ...
1
vote
1answer
40 views

Primary ideal exercise

I have an exercise about the properties of primary ideal. It's Exercise 15.17 of "Step in commutative algebra", R. Y. Sharp. Let $(A,\mathfrak{m})$ be a local ring and $I$ be a proper ideal of $A$. ...
0
votes
0answers
21 views

references for number ring theory [on hold]

I am currently studying commutative algebra and in most ressources I have found, I am quite unhappy with the part devoted to the study of "standard" examples, and I find difficult to get surveys that ...
1
vote
0answers
34 views

A query on Veronese mapping

The Veronese mapping defined as usual on some $P^n$. Then it is certainly regular. I want to prove that the inverse map to this map is also regular. I have an idea to use projections with ...
13
votes
5answers
551 views

Irreducibility of Polynomials in $k[x,y]$

I'm working through some Hartshorne problems and have noticed that in order to do certain problems properly one must prove a given polynomial $f\in k[x,y]$ is irreducible. For example, in problem ...
0
votes
0answers
27 views

Colon ideal and Artin-Rees lemma [on hold]

Let $a$ and $b$ be ideals of a Noetherian integral domain $R$. Prove that there exists a natural number $r$ such that $(a^n:b) = a^{n-r}(a^r:b)$ for $n > r$.
1
vote
1answer
48 views

If $p \in \operatorname{Ass}M$, then $R/P \subset M$.

Let $R$ be a commutative ring with unity. $M$ an $R$-module. Then $P \in \operatorname{Ass}M$ if and only if there is a submodule $N\subset M$ such that $R/P \cong N$. ...
1
vote
2answers
22 views

$rad(I)=\cap_{I\subset P,~P~prime}P$

$R$ commutative ring with unity. $I$ R-ideal. Then $rad(I)=\cap_{I\subset P,~P~prime}P$. That is, the radical of $I$ is the intersection of all prime ideals containing $I$. There is a proof of this ...
1
vote
2answers
25 views

minimal prime ideals over the union of two prime ideals

When two subvarieties intersect properly ($X_1\cap X_2$), it should end up with a new subvariety($X_3$=$X_1\cap X_2$). I do not know how to keep track of the intersection operation from the algebraic ...
9
votes
2answers
788 views

Tensor product of domains is a domain

I'm reading Milne's Algebraic Geometry course notes, version 5.22, as a companion to an algebraic geometry course I'm taking now. Proposition 4.15 states: Let $A$ and $B$ be $k$-algebras, which are ...
1
vote
0answers
182 views

Multiplicity and regular sequences

We define multiplicity of a module $M$ of dimension $d>0$ as $$e(M) := \operatorname{lc} (P_M) (d-1)!,$$ where $P_M$ denotes the Hilbert polynomial of $M$ and $\operatorname{lc}(P_M)$ its leading ...
4
votes
1answer
423 views

Generators for the intersection of two ideals

Let $I=\langle a_1,\dots, a_s\rangle, J=\langle b_1,\dots, b_t\rangle$ be ideals of arbitrary commutative ring. Then we know that $I+J=\langle a_1,\dots, a_s, b_1,\dots, b_t\rangle, ...
1
vote
1answer
26 views

“Adjugate” of an endomorphism of a finite-rank free module

If $M$ is a free module of finite rank $n$ over a commutative unitary ring and $a$ is an endomorphism of $M$, consider the endomorphism $\hat a$ of $M$ defined by the identity $$ x_1\wedge ...
9
votes
2answers
422 views

Vanishing of a certain Tor

I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes and there are some commutative algebra facts that I am not able to figure out by myself apparently, ...
0
votes
1answer
36 views

A question related to the height of a proper ideal in a Noetherian ring

Let $A$ be a Noetherian ring, and $I\subset A$ a proper ideal of height $r$. Is it true that there exist $a_1,\ldots,a_r\in I$ such that $$\operatorname{ht}(a_1,\ldots,a_j)=j$$ for all $j=1,\ldots,r$ ...
0
votes
1answer
52 views

Relation between ideals in Noetherian domains.

Suppose that we have a Noetherian domain $R$ and two ideals $I$ and $J$ of $R.$ Now consider the minimal (or irredundant) primary decompositions $I=\bigcap\limits_{i=1}^r Q_i$ and ...
2
votes
0answers
44 views

Generators of an ideal in rings of power series

Please help me for solving a homework. Let $k$ be a field and $R=k[[x_1,x_2,\ldots,x_n]]$ the ring of power series over $k$. If $I$ is an ideal of $R$ such that ...
0
votes
0answers
29 views

The relation of colon ideal and quotient [on hold]

Let $k$ be a field and $R=k[[x_1,x_2,\ldots,x_n]]$ the ring of power series over $k$. If $I$ is an ideal of $R$ such that the cardinal of the set $\{ann(f+I): f \in R \}$ is two, what can we say ...
3
votes
1answer
279 views

Example of a Noetherian module that is not a ring?

I've been trying to think of an $R$-module that is Noetherian, not finite and is not a ring. Examples that I know are: 1 A finite Abelian group is a Noetherian $\mathbb Z$-module (of course it ...
3
votes
1answer
27 views

Basis-free and noncommutative versions of the two-polynomials-over-ring problem (McCoy theorem etc.)

There is a rather canonical bunch of exercises in commutative algebra which tend to come up time and again on math.stackexchange: recently in #948010 and #83121, formerly in #227787 and #413788, and ...
8
votes
1answer
227 views

Irreducible polynomial over an algebraically closed field

Suppose $k$ is an algebraically closed field and $p(x,y)\in k[x,y]$ is an irreducible polynomial. Prove that there are only finite many $a\in k$ such that $p(x,y)+a$ is reducible, i.e. the set ...
0
votes
0answers
30 views

Differential operators on the polynomial ring

Let $A$ be a commutative algebra over complex numbers. If $a\in A$ we define $m_a$ to be a linear map which sends each $x$ to $ax$. The zero map $A\to A$ is said to be a differential operator of an ...
0
votes
1answer
39 views

Question about the ideal $I=(xy,yz,zx)$ in the ring $\mathbb C[x,y,z]$.

Given the ideal $I=(xy,yz,zx)$ in the ring $\mathbb C[x,y,z]$, I want to compute $V(I)$, which is the intersection of all ideals containing $I$. And I also want to prove that $I$ can't be generated by ...
2
votes
2answers
87 views

Showing an ideal with maximality condition is prime.

Let $R$ be a commutative domain and suppose that $I \subseteq R$ is an ideal of $R$ maximal with respect to the property that $I^{-1} \not\subseteq R$. Show that $I$ is a prime ideal. This is ...
2
votes
0answers
84 views

Computing generators of the positive component of a graded ring

Let $R$ be a sub-algebra of $\mathbb{Q}[X_1^{\pm 1}, \dots, X_n^{\pm 1}]$ given by finitely many generators, and let $\lambda$ be a linear form $\lambda : \mathbb{Z}^n \to \mathbb{Z}$. This defines a ...
1
vote
0answers
28 views

K[x,y,z,w]/(xw-yz) not UFD [duplicate]

I am trying to prove its not UFD. I started by assuming x=ab in K[v] , where v=v(xw-yz) then x-ab=(xw-yz)f, for some f in K[x,y,w,z] I tried to say that deg of a, and b is less or equal 1, and ...
0
votes
1answer
34 views

Cohen structure theorem for artinian local rings

Let $(R,m)$ be an artinian local ring. Since $m^n=0$ for some $n$, it is clear that $R$ is complete with respect to $m$-adic topology. Now i want to know that how do we state the Cohen structure ...
2
votes
1answer
57 views

Sets of prime ideal contain a minimal element

I want to prove that every nonempty set of prime ideal contain a minimal element, my attempt is to prove it by using zorns lemma and i would like to know if my proof is valid. Let $\Sigma$ be a ...
5
votes
1answer
91 views

A certain valuation of $k(X,Y)$ with value group $\mathbb{Z}+\mathbb{Z}\alpha$

Let $k$ be a field, $X$ and $Y$ indeterminates, and suppose that $\alpha$ is a positive irrational number. Then the map $\nu:k[X,Y]\rightarrow \mathbb{R}\cup \{\infty\}$ defined by $\nu\left(\sum ...
1
vote
1answer
64 views

Intersection of two polynomial ideals

In the $4$-dimensional affine space $\mathbb{A}^4$ with coordinates $x,y,z,t$, consider $X$ as the union of the planes $$ X'=\{x=y=0\} $$ and $$ X''=\{z=x-t=0\} $$ (I'm working on a algebraically ...
0
votes
1answer
27 views

maximal algebraically independent sets in ring extensions

Let $E/K$ be a field extension. It is a well known fact that all maximal subsets $A \subset E$ consisting of algebraically independent elements over $K$ have the same cardinality (which is by ...
-3
votes
1answer
117 views

Lexicographic monomial orders [closed]

Is it true that in $k[x,y]$ the monomial orders deglex and degrevlex are same? Here, deglex and degrevlex are defined as in Sage: degrevlex: Degree reverse lexicographic deglex: Degree ...
1
vote
1answer
48 views

Krull dimension in finite ring extensions

Let $K$ be a field and $R=K[a_1, \dots, a_n]$ a finite ring extension. Suppose that the degree of transcendence of $R$ over $K$ is $r$. Then the Krull dimension of $R$ is at most $r$. I would like to ...
3
votes
1answer
73 views

Localization at finitely many minimal prime ideals

Let $A$ be a commutative ring with finitely many minimal prime ideals $\{p_1,\dots,p_n\}$. Let $A_{p_1,\dots,p_n}$ be the localization of $A$ away from the minimal primes, i.e. $S^{-1}A$ where $S = ...
0
votes
0answers
32 views

Bruns-Herzog, Cohen-Macaulay Rings, Exercise 9.1.10(c)

The following question is from the book: Bruns-Herzog, Cohen-Macaulay Rings, Exercise 9.1.10(c). Let R be a ring, I a finite generated ideal and M an R-module. Let R$_{\infty}$ be a polynomial ...
1
vote
0answers
94 views

Learning roadmap in Algebra

I am a senior undergraduate student in mathematics, I have a sound knowledge in the following areas: a) Commutative Algebra b) Field Theory and Galois Theory c) Homological Algebra My question is ...
-1
votes
1answer
60 views

What are good references for the theory of Cohen-Macaulay rings?

I am studying Cohen-Macaulay Rings from the book Cohen-Macaulay rings by W. Bruns. Please tell me some reference book/notes on Cohen-Macaulay rings theory.
4
votes
3answers
82 views

stalk of projective variety in terms of the coordinate ring

Let $X \subseteq \mathbb{P}^n$ be an embedded projective variety over some field $k$ with its corresponding homogeneous coordinate ring $R = k[X_0,\dots,X_n]/I(X)$. Let further $X = \bigcup_{i=0}^n ...
1
vote
1answer
38 views

Colon ideal of fractional ideals is itself a fractional ideal

I received this question on homework in my homological algebra class and I need some guidance. Let $R$ be a commutative integral domain and $K$ be its field of fractions. A fractional ideal $I$ of ...
0
votes
1answer
68 views

Radical ideals of $\mathbb{Z}$?

I am having trouble with classification of the radical ideals of $\mathbb{Z}$. We know that for a commutative ring $R$ with an ideal $I$, the radical of $I$ is defined (and denoted as $\sqrt{I}$) as ...
0
votes
1answer
71 views

Is $\mathbb Z\oplus \mathbb Q$ a flat $\mathbb Z$-module? [closed]

Can someone please explain why $\mathbb Z\oplus \mathbb Q$ is flat or not?
5
votes
1answer
68 views

How to show that $\mathbb{C}[x_1,x_2,x_3, x_4]/(x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is integral domain

I am looking for a way to show that the ring $\mathbb{C}[x_1,x_2,x_3, x_4]/I$ where $I = (x_1x_2 - x_4x_3, x_1x_3 - x_2x_4, x_4x_1 - x_3x_2)$ is an integral domain. In other words I want to show $I$ ...
-1
votes
0answers
30 views

Relation between elements of a ring and their annihilators

let $(R.m)$ be a local ring and $x,y$ two elements of $R$ and for ideal $I$ of $R$, we have $x$ is in $I$, $ann(x)=ann(y)$ and $x$ is uniqu minimal ideal of $R$, is there any conditions that implies, ...
2
votes
0answers
37 views

Category of schemes with flat morphisms

Consider the category whose objects are schemes, and for every two schemes $X$ and $Y$, morphisms $\operatorname{Hom}(X,Y)$ consist of flat morphisms $X\rightarrow Y$, only. Does this category have a ...
8
votes
1answer
167 views

Recovering free modules from their projective limit

Let $\dotsc A_2 \to A_1 \to A_0$ be a sequence of surjective homomorphisms of commutative rings. Consider the projective limit $\varprojlim_i A_i$. If $S$ is an (infinite) set, then $\varprojlim_i ...
15
votes
2answers
167 views

Is Pythagoras the only relation to hold between $\cos$ and $\sin$?

Pythagoras says that $\cos^2 \theta + \mathrm{sin}^2\theta = 1$ for all real $\theta$. (Vague) Question. Is this the only relationship between the functions $\cos$ and $\sin$? More precisely: Let ...
2
votes
0answers
54 views

Is the maximal ideal of a localization at a prime ideal principal?

Let $X$ be a closed subvariety of $\mathbf P^{n}_{k}$ which is nonsingular in codimension one. Let $Y$ be a subvariety of $X$ of codimension one, let $\eta$ be its generic point. First question: is ...
1
vote
2answers
129 views

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 ...