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

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About the Linear Quotients of Square of an Ideal with Linear Quotients

Let $I$ be a monomial ideal generated by quadratic monomials $u_{1},...,u_{s}$ and suppose that $I$ has linear quotients with respect to this given ordering. Is it true or false that $I^{2}$ has ...
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4answers
374 views

Let $R$ be a commutative ring with $1$. Suppose that every nonzero proper ideal of $R$ is maximal. Prove that there are at most two such ideals.

Let $R$ be a commutative ring with $1$. Suppose that every nonzero proper ideal of $R$ is maximal. Prove that there are at most two such ideals. Help me some hints. I have no idea to start. ...
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1answer
67 views

$k[[x]]$-modules.

Let $k$ a field with $\bar{k}=k$. What can be said about the ideals of $k[[x]]$, can they be determined? I am looking for a $k[[x]]$-module $S$ such that the map $s\mapsto xs$ is surjective, does such ...
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2answers
141 views

Prime ideals in a principal ideal ring

I know that in a principal ideal DOMAIN every $\neq 0$ prime ideal is maximal. is this also true for just a commutative principal ideal ring? It seems to be true for $\mathbf{Z}/n\mathbf{Z}$ ($>1$, ...
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2answers
309 views

A commutative ring whose all proper ideals are prime is a field. [closed]

Let $R$ be a commutative ring with $1$. Suppose that all ideals $I \neq R$ are prime. Prove that $R$ is a field. Help me some hints. Thanks a lot.
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1answer
183 views

Saturation of a Graded Radical Ideal in $S=k[x_{1},…,x_{n}]$

Let $I\subset S=k[x_{1},...,x_{n}]$ be a graded radical ideal different from $\mathfrak{m}=(x_{1},...,x_{n})$. Prove that $I$ is saturated. To prove that $I$ is saturated it is sufficient to ...
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1answer
563 views

Krull dimension of some quotient rings

I have difficulties in doing some calculations of heights and Krull dimensions; I hope that somebody could help me unveil the "tricks of the trade". In the following $\alpha,\beta,\gamma$ denote ...
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1answer
305 views

Examples of non-isomorphic fields with isomorphic group of units and additive group structure

YACP mentions in a comment that: There are examples of non-isomorphic fields $K$ and $L$ with $(K,+)\cong (L,+)$ and $(K^{\times} ,\cdot)\cong (L^{\times},\cdot)$ Can someone provide an ...
3
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1answer
71 views

Finite intersection of DVRs

Let $K$ be a field and $R_1,\dots,R_n$ DVRs of $K$ with $m_i$ the maximal ideal of $R_i$ and $R_i \not\subseteq R_j$ for $j\neq i$ . Define $A=\bigcap_{i=1}^n R_i$. Then $A$ is semilocal with maximal ...
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1answer
62 views

All maximal ideals in the ring of polynomials of are of the kind $N_p=\langle x_i-p_i:i=\overline {1,n}\rangle$ for some point p in the affine space

I am reading a proof on the coincidence of the functional field of a variety (defined by equivalence classes of regular functions) and the field of quotients of its coordinate ring. It turns out I ...
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2answers
250 views

Structure of finitely generated modules over local Artinian rings

Let $(R,m)$ be an Artinian local ring with $m^2=0$. Let $M$ be a finitely generated $R$ module. Can we say anything about the structure of $M$? Perhaps to give a complete structure might be very ...
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2answers
184 views

Finite injective dimension

Let $A$ be a commutative noetherian ring. Is it true that if $A$ is regular then any module over it has a finite injective dimension? What if $A$ is Gorenstein? Any reference who discuss this?
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1answer
106 views

Conditions on $a,b\in\mathbb{Q}$, for $a+b\sqrt{n}$ to be integral over $\mathbb{Z}$

For $n\in \mathbb{Z}$ square-free, let $$k:=\mathbb{Q}(\sqrt{n}),$$ and let $$\alpha:=a+b \sqrt{n}\in k.$$ Prove that $$ \alpha \mbox{ is integral over } \mathbb{Z}\;\;\; \Longleftrightarrow ...
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1answer
113 views

Maximal regular sequences of different length

This question is Exercise 1.2.20 in the book: Winfried Bruns, H. Jürgen Herzog, Cohen-Macaulay Rings, Cambridge University Press, 1998. Let $k$ be a field and $R=k[[X]][Y]$. Deduce that $X, Y$ and ...
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3answers
633 views

Verifying that the ideal $(x^3-y^2)$ is prime

How to prove that the ideal $I=(x^3-y^2)$ in $k[x,y]$ is prime? I have constructed a map from $k[x,y]$ to $k[t]$, which maps $x$ to $t^2$, and $y$ to $t^3$. Then, I want to show that the kernel ...
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80 views

Give an infinite sequence of principal ideals of $R$ such that the ascending chain condition does not hold

Let $R=\{\sum_{i=0}^n a_ix^i\mid n\geq 0, a_0\in\mathbb{Z}, a_i\in\mathbb{Q} \text{ for } i\geq 1\}$. Give an infinite sequence of principal ideals of $R$ such that the ascending chain condition ...
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2answers
139 views

Taylor series in algebraic geometry

Let $F\in k[T_1,\ldots, T_N]$ be a non-zero polynomial and take $x=(x_1,\ldots,x_N) \in \mathbb A^N$. Then $F$ has an expression in Taylor series in $x$ $$F(T)=F^{(0)}(T)+F^{(1)}(T)+\cdots ...
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1answer
207 views

Generalization of Cayley-Hamilton

I'm having trouble following a proof of this generalization of the Cayley-Hamilton theorem: Suppose that $M$ is an $A$-module generated by $n$ elements, and that $\varphi \in ...
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1answer
109 views

Minimal Ideal of a Commutative Ring with Unity

Can anyone help me prove this? This one is from Malik's Fundamentals of Abstract Algebra. An ideal $I$ of a ring $R$ is called a minimal ideal if $I≠{0}$ and there does not exist any ideal $J$ of R ...
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70 views

Spectral sequences to involve together two ideals of a ring

I'm looking for spectral sequences to involve together two ideals of a ring. For instance, let $I,J$ be two ideals of Noetherian ring $R$ and $M$ be a finite $R$-module then we have the following ...
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1answer
141 views

Hartshorne's connectedness theorem

A key step in proving Hartshorne's connectedness theorem is the following: Let $(R,\mathfrak m)$ be a Noetherian local ring such that $\operatorname{depth}R\geq 2$. Then ...
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1answer
63 views

Find the support of a module and associated primes

I have the module $M=k[x,y]/(x+y)$ and I must find the sets Supp(M) and Ass(M). Could anyone give me a tip how to make it. I have no idea about how to make it. Thank you...
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Proving integrality of the coefficients “inside the box”

Consider the (usual) $ABKL$ setting: $A$ is an integral domain with field of fractions $K$, $L/K$ is an algebraic field extension, and $B$ is the integral closure of $A$ in $L$ (we are not assuming ...
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1answer
70 views

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 ...
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1answer
50 views

regarding a certain homomorphism of finite free modules

Consider the following situation: let $(R,m)$ be a local Noetherian ring and $f: F_1 \rightarrow F_0$ a morphism of finite free $R$-modules with $\operatorname{rank}(F_i) = r_i, \, i=0,1$. Suppose ...
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2answers
134 views

Nakayama's Lemma

When we prove Nakayama's Lemma, which states that if $M$ is a finitely generated $R$-module, where $R$ is a commutative ring and if $I$ is an ideal of $R$ contained in the Jacobson radical of $R$, if ...
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1answer
108 views

Pro$-p-$group as a $\Lambda-$module

Let $p$ be a prime number, and let $X$ be an abelian pro$-p-$group (i.e for some indexing set $I,$ we have $X=\varprojlim X_i$ where $X_i$ is a finite, abelian $p-$group for each index $i \in I .$) ...
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1answer
171 views

Associated graded ring and completion

Let $ R $ be a ring and $I$ an ideal. This gives a decreasing filtration of $R$ by $ I^n$. Consider the associated graded ring $ \operatorname{gr}_I(R) =\bigoplus I^n/I^{n+1}$. As it is graded, there ...
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1answer
60 views

Bruns and Herzog Proposition 1.4.11

Proposition 1.4.11 in Bruns and Herzog Cohen-Macaulay Rings reads: Proposition: Let $R$ be a Noetherian ring and $\phi: F \rightarrow G$ a homomorphism of finite free $R$-modules. Then $rank ...
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2answers
160 views

Associated primes of a quotient module.

Let $R$ be a Noetherian ring, $M$ a finitely generated $R$-module and $p\in \operatorname{Ass}(M)$. Suppose $x$ is an $M$-regular element and $q$ is a minimal prime over $I=(p,x)$. How can we show ...
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1answer
77 views

Inclusion containing V and radicals

Let $A$ be a commutative ring with unit and $I,J$ be two ideals of $A$. Also, denote $V(I):=\{\mathfrak{p}\in\operatorname{Spec}A\mid I\subset\mathfrak{p}\}$. Why is it true that if $J\subseteq ...
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1answer
98 views

A finite unital and commutative ring with exactly one maximal ideal has $p^{n}$ elements.

Suppose $R$ is a finite unital and commutative ring that has exactly one maximal ideal. Prove that $\left | R \right |=p^{n}$ where $p$ is a prime number. If $R$ will be non-commutative, do we have ...
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2answers
61 views

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 ...
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79 views

Prove that these are primary decompositions.

Prove that $$\langle 4,2x,x^{2} \rangle=\langle 4,x \rangle\cap \langle 2,x^{2} \rangle $$ $$\langle 9,3x+3 \rangle=\langle 3 \rangle\cap \langle 9,x+1\rangle$$ are two primary decomposition in ...
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3answers
173 views

Show that $M[x]$ is a Noetherian $A[x]$-module.

This is a question from Atiyah and Macdonald, Introduction to Commutative Algebra. Problem: Let $M$ be a Noetherian $A$-module. Show that $M[x]$ is a Noetherian $A[x]$-module. Solution: So, I ...
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1answer
124 views

Radical of ideal

I am trying to compute the radical of ideal $I=((X-Z)(X-Y)(X-2Z),X^2-Y^2Z)$. I suspect that rad$(I)= (X-Z) \cap (X-Y) \cap (X-2Z) \cap (X^2-Y^2Z)$ which is $(X-Z)(X-Y)(X-2Z)(X^2-Y^2Z)$. Is my ...
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1answer
131 views

Ring epimorphism $f:R\rightarrow S$, $R$ has finitely many maximal ideals, then $f(J(R))=J(S)$.

Suppose $R$ and $S$ are commutative rings with unit, and $f:R\rightarrow S$ is an epimorphism. Prove that: $$f(J(R))\subseteq J(S).$$ If $R$ has finitely many maximal ideals, then prove that: ...
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96 views

Hilbert series of the polynomial ring $K[X_1, \dots, X_s]$

Let $K$ be a field and $a_1, \dots, a_s \in \mathbb{N} \setminus \{0\}$. How can I compute the Hilbert series for $K[X_1, \dots, X_s]$, where $\deg(X_i)=a_i$?
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164 views

Induced map on spectra of rings

Let $B$ be a ring containing $A$, and the ring extension is integral. Furthermore, $B$ is a finitely generated $A$-algebra. Then how to show that the induced map on spectra of the rings is a finite ...
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1answer
83 views

Write $\mathbb{R}[x]/(x^5+x^3)$ as direct product of its localizations

Let's consider the commutative ring $\mathbb{R}[x]/(x^5+x^3)$. We have that $x^5+x^3=x^3(x^2+1)$. So $\mathbb{R}[x]/(x^3(x^2+1)) \simeq \mathbb{C}[x]/(x^3) $. How can I write the artinian ring ...
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1answer
138 views

Dimension of a curve

I'm trying to understand this example: Let $f(T_1,T_2)\subset k[T_1,T_2]$ be a non-constant irreducible polynomial. Let $X=Z(f)\subset \mathbb A^2$. We will see that $\dim(X)=1$. We have ...
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2answers
86 views

If $X=\{x\}$, then $\dim(X)=0$

If $X$ is a quasiprojective variety, then by definition $\dim(X)=trdeg(k(X)|k)$. I'm trying to prove if $X=\{x\}$ is a point, then $\dim(X)=0$ I'm already proved that $k[X]\cong k$, now if I prove ...
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1answer
155 views

Conductor of a ring extension

Consider the rings $A:=\mathbb{C}[x,y]-\mathbb{C}^{\times}x$ and $B:=\mathbb{C}[x,y]$. I am working on something where I need to find a relation between $A/\mathfrak{c}$ and $B/\mathfrak{c}$, ...
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1answer
107 views

Help in this easy lemma about dimension in algebraic geometry

I'm studying dimension of quasiprojective varieties which is defined as $$\dim(X)=trdeg(k(X)|k)$$ if $X$ is a quasiprojective variety. I didn't understood this lemma: If $f:X\to Y$ is a finite ...
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1answer
64 views

Why $k(\mathbb A^n)=k(T_1,\ldots,T_n)$?

The book I'm reading defines the dimension of a quasi-projective variety $X$ as $$\dim (X)=trdeg(k(X)|k)$$ Since $K(X)$ is a field containing $k$, dimension is well-defined. In order to prove that ...
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1answer
218 views

An example of ideal that has no primary decomposition.

Give an example of a commutative ring with unit and an ideal that has no primary decomposition. I think boolean Ring will be the right example, but I don't know how I must show that. So please ...
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1answer
52 views

The height of a prime ideal in the $\kappa[[X]][Y]$

Let $\kappa$ be a field and $S=\kappa[[X]]$ be the ring of power series which depends on the indeterminate $X$. Now consider the ring $S[Y]$, the ring of polynomials with coefficients in $S$ and ...
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1answer
143 views

Ext of an $\mathfrak{m}$-primary ideal

Let $(A,\mathfrak m,k)$ be a Noetherian local ring, $M$ a finitely generated $A$-module, and $I$ an $\mathfrak{m}$-primary ideal. If $\operatorname{Ext}^{i}_{A}(A/\mathfrak{m},M)=0$ then ...
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1answer
77 views

Localization of two rings which is an integral extension, then integral extension still holds?

Question seems simple, but I just can't find the solution. Let $A/B$ be an integral ring extension and let $P$ be a prime ideal of $B$. By lying-over theorem, there is $Q$, a prime ideal of $A$, ...
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1answer
58 views

$\mathbb Z_p[T]/(T^a,p^b)\cong\mathbb Z_p[[T]]/(T^a,p^b)$

Let $a,b\in \mathbb N,$ then $$\mathbb Z_p[T]/(T^a,p^b)\cong\mathbb Z_p[[T]]/(T^a,p^b)$$ 1.What is this isomorphism ? 2.How to prove that $|\mathbb Z_p[[T]]/(T,p)^t|=p^{t(t+1)/2}$ Now ...