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

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2answers
101 views

isomorphic ideals and projective dimensions of quotients

Let $R$ be a Noetherian ring and $I,J$ proper ideals that are isomorphic as $R$-modules. Can we conclude that the projective dimensions of $R/I,R/J$ are equal?
0
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2answers
66 views

projective dimension and localization

Let $R$ be a Noetherian ring and $M$ a non-zero finite $R$-module with finite projective dimension equal to $n$. For any $P$ inside the support of $M$ we have that $\operatorname{projdim} M_P \le ...
4
votes
1answer
36 views

Is $S(E\otimes_AB)\cong S(E)\otimes_AB$?

Suppose A is a commutative ring, $E$ is an $A$-module, $B$ is an $A$-algebra, ${S}$ is the symmetric $A$-algebra functor. Is $S(E\otimes_AB)\cong S(E)\otimes_AB$? I try to use universial property, ...
8
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2answers
127 views

$R/I$ is not Noetherian. Prove that $I$ is a prime ideal.

Let $R$ be a commutative ring with $1$ and let $I$ be an ideal of $R$, maximal with respect to the property that $R/I$ is not Noetherian. Prove that $I$ is a prime ideal. I need some hints to ...
3
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1answer
110 views

$B$ is a finitely generated module over $A$. Then if $A$ is a Noetherian ring (or Artinian ring), then $B$ is a Noetherian (Artinian) ring.

Let $A$ be a subring of $B$. Suppose $B$ is a finitely generated module over $A$. Prove that if $A$ is a Noetherian ring (or Artinian ring), then $B$ is a Noetherian (Artinian) ring. I am quite ...
0
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1answer
55 views

Find the spec of a localized ring.

I must find the following: $$\operatorname{Spec}\left(\Bbbk[x,y]/\left<xy-1\right>\right)$$ Is there a way to describe that set? I am trying to find the laurent polynomials rings prime ideals. ...
3
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2answers
220 views

Localization of an integral domain and fields of fractions

Is it true that every localization of an integral domain is isomorphic to a subring of its field of fractions? How are the localizations of an integral domain related to its field of fractions? Is ...
2
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1answer
209 views

Primary decomposition of an ideal

Suppose $I$ is the ideal $(xy, yz, zx)$ in $R =\Bbb R[x, y, z]$. I want to compute the primary decomposition of $I$. I have viewed many post on this topic, as I suspect, the primary ...
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1answer
121 views

A property of $I$-adic topologies

Let $R$ be a commutative ring with multiplicative identity and $I$ a proper ideal of $R$ such that the intersection of its powers is the zero ideal. It can be shown that if the $I$-adic topology is ...
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0answers
76 views

$V(f)$ is irreducible iff $f=g^k$, $g$ irreducible

I'm trying to prove this theorem $V(f)$ is irreducible iff $f=g^k$, $g$ irreducible. To prove the converse, we have $V(f)=V(g^k)=V(g)$, since $g$ is irreducible $V(g)$ is irreducible, then ...
2
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2answers
85 views

$(R_1\oplus R_2) [x]/(p(x)) = R_1[x]/(p(x))\oplus R_2[x]/(p(x))$?

For convenience, I shall use '$=$' to denote isomorphisms. Suppose we have a commutative ring $R = R_1\oplus R_2$, and $(p(x))$ is the ideal generated by $p(x)\in R[x]$. Can we deduce that ...
1
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1answer
51 views

Criteria for local freeness of a module [duplicate]

Let $M$ be a finite type module over a local ring $R$. If the minimum number of generators equals the maximum number of independent elements, is $M$ free? If not, do you have a counterexample?
4
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0answers
227 views

Irreducible homogeneous ideals

I have the following question: Let $I$ be a homogeneous ideal. Is it true that $I$ is irreducible if and only if it can't be written as the intersection of two homogeneous ideals? So, is it ...
2
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1answer
71 views

Is weak Hilbert Nullstellensatz's theorem “if and only if”?

My question is quite simple, I would like to know if the weak Hilbert Nullstellensatz's theorem can be "if and only if": Nullstellensatz theorem If $I\subset k[X_1,\ldots,X_n]$ is a proper ideal, ...
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3answers
601 views

Integral domain whose irreducible elements are not prime

Is there some integral domain such that none of its irreducible elements is prime? Recall that a nonzero, non invertible element $a$ of an integral domain $D$ is said to be Irreducible, if for ...
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1answer
47 views

degree zero term of minimal free resolution

Let $R=k[x_{1},\ldots,x_{n}]$ where $k$ is a field, and let $I$ be a homogenous ideal. Suppose that $\cdots\to R_{1}\to R_{0}\to R/I\to 0$ is a (the) graded minimal free resolution of $R/I$. Is it ...
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0answers
98 views

What is $Spec(\mathbb{Z}[x])$? [duplicate]

What is $Spec(\mathbb{Z}[x])$? For a commutative ring $A$ e with $1$, its spectrum $Spec(A)$ is defined to be the set of all of its prime ideals. So the question is to find all the prime ideals of the ...
3
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1answer
281 views

Primary decomposition of a monomial ideal

Can anyone give me an idea about the primary decomposition of the ideal $I=(x^3y,xy^4)$ of the ring $R=k[x,y]$? I am trying to connect the primary decomposition with the set Ass(R/I) which i ...
2
votes
4answers
213 views

Consider $R[x]$ and let $S$ be the subring generated by $rx$, where $r \in R$ is some non-invertible element. Then $x$ is not integral over $S$

Consider $R[x]$ and let $S$ be the subring generated by $rx$, where $r \in R$ is some non-invertible element. Then I want to show that $x$ is not integral over $S$ I'm not seeing why this is the ...
0
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1answer
127 views

Show that either $x$ or $1-x$ is invertible in $R$ [closed]

Let $F$ be a field, and let $R$ be a subring of $F$. Suppose that for each $u\in F\setminus \{0\}$, either $u\in R$ or $u^{-1}\in R$. Given $x\in R$, show that either $x$ or $1-x$ is invertible (or ...
4
votes
2answers
348 views

Module homomorphism surjective but not injective

It's a well-known exercise in commutative algebra to show that if an A-module endomorphism of a noetherian module M is surjective, it's also injective. What are examples that the statement is wrong ...
0
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1answer
37 views

Integral dependence relation

Let $K$ be a field. Then $K[X^2]$ in contained in $K[X]$, and it is a finite ring extension. Now let $P(X)$ be a polynomial of $K[X]$. What is the polynomial that $P(X)$ satisfies over $K[X^2]$? Can ...
3
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1answer
84 views

A criterion for an extension to be Galois

This is an exercise given during my Commutative Algebra course. I reached to solve just the "if" arrow, but not the "only if". The question is: Let $F\subseteq L$ be a finite degree extension of ...
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0answers
87 views

$f$ is irreducible iff $V(f)$ is irreducible

I would like to know if the following statement is true: $f$ is irreducible iff $V(f)$ is irreducible. My tools I'm trying to use to prove this are Study's Lemma and basic algebra. If $f$ is ...
0
votes
1answer
220 views

Jacobson radical of a finite commutative ring [closed]

Let $R$ be a finite commutative ring, and let $J$ be the Jacobson radical of R (the intersection of all the maximal ideals of R). (1) Prove that $J^n=0$ for some $n$. (2) Use the Chinese ...
4
votes
2answers
141 views

Is an ideal which is maximal with respect to the property that it consists of zero divisors necessarily prime?

This is in follow-up to this question. Let $R$ be a commutative ring with identity and consider the set $Z \subset R$ of zero divisors. If the ideal $I\subset Z$ is maximal with respect to the ...
0
votes
1answer
77 views

Fundamental questions on rings of polynomials.

Put $\mathfrak{E}$ the union of $(0,0)$ and $k\times 1$ in $k^2$ ($k$ an algebraically closed field). Furthermore let $\mathfrak{Z}$ the set of all $f\in k[x,y]$ such that $f(s)=0$ for all ...
3
votes
0answers
57 views

“Localization” of a module at a family of elements

Let $x=(x_i)_{i \in I}$ be a family of elements of a commutative ring $R$. Typically $I$ is infinite. Let $M$ be an $R$-module. For every finite subset $E \subseteq I$ define $M_E = M$, and for ...
2
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1answer
87 views

Intersection of a PID and a field

There is an exercise in Bourbaki about the intersection $A = k (x,y) [z] \cap k (z, x + yz)$. These are two subrings of $k (x,y,z)$. The first is PID, the second is a field. Bourbaki requests to prove ...
3
votes
1answer
106 views

Saturation and Associated Primes of an Ideal

If $I$ is an ideal of a Noetherian ring $S$ and $x,y\in S$, show that the following are equivalent: $(1)$ $(I:y^{\infty})=(I:(x,y)^{\infty})$ $(2)$ Every associated prime of $I$ that ...
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0answers
43 views

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 ...
8
votes
4answers
365 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. ...
1
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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 ...
1
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2answers
138 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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1answer
117 views

About Saturation of Monomial Ideals [closed]

Let $I\subset S=K[x_{1},...x_{n}]$ be a monomial ideal and let $J=(x_{1},...,x_{r}).$ Show that $I:J\neq I$ if there exist integers $a_{i}>0$ such that $x_{i}^{a_{i}}\in G(I)$ for $i=1,...,r.$
1
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2answers
291 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.
3
votes
1answer
170 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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votes
1answer
533 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 ...
15
votes
1answer
289 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
69 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
61 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 ...
2
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2answers
232 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 ...
2
votes
2answers
176 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
102 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 ...
6
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3answers
575 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 ...
1
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0answers
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 ...
1
vote
2answers
135 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 ...
5
votes
1answer
201 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 ...
4
votes
1answer
100 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 ...