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

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1answer
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If $R/I$ satisfies Serre's condition $S_2$ then $R/(I,x)$ satisfies $S_1$

Let $I$ be an ideal of polynomial ring $R=K[x_1,\ldots,x_n]$ and $x$ be a non-zero divisor of $R/I$. Is the following statement true? If $R/I$ satisfies Serre's condition $S_2$ then $R/(I,x)$ ...
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1answer
58 views

What is $\overline{\{ x \}}$ in Atiyah-Macdonald?

On pg. 13 of Atiyah-Macdonald's "Introduction to Commutative Algebra": 18.ii) Prove that $\overline{\{ x \}}=V(p_x)$ What is $\overline{\{ x \}}$? Is it the closure of prime ideal $x$? I assumed ...
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2answers
69 views

Noetherian Jacobson rings

One of the equivalent forms in definition of a Noetherian Jacobson ring $R$ is that $R$ has no prime ideals $P$ such that $R/P$ is a 1-dimensional semi-local ring. When $R/P$ has dimension 1, it ...
3
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1answer
38 views

Rings of algebraic integers

A basic question on algebraic numbers. If $L/K$ is a finite extension of number fields with respective rings of integers $\mathcal O_L$ and $\mathcal O_K$ then is it true that $\mathcal O_L$ is ...
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1answer
89 views

Morphism of rings and localization

Let $ \varphi : A \to B $ be a morphism of rings. Why are the two following assertions equivalent: $ 1) $ There exists a multiplicative subset $ S $ of the ring $ A $, and an ideal $ I $ of $ A $, ...
2
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1answer
72 views

what are the “points” of the scheme $\mathbb{Z}_8[x] /(x^2 + 7)$

I noticed modulo 8 the quadratic $x^2 + 7$ is zero for four separate values $x = 1,3,5,7 \in \mathbb{Z}_8$. The number of zeros exceeds the degree. I would like to define the "variety" ...
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1answer
43 views

A local PID is a Euclidean domain

Studying commutative algebra I've encountered this statement: A PID which is also a local ring is a Euclidean domain. Is it true ? Why ?
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1answer
37 views

A condition for a homogeneous ideal to be prime

The following is the problem 11 of Chaper 8 Section 4 of Ideals, Varieties, and Algorithms by Cox, Little and O'Shea. A homogeneous ideal is said to be prime if it is prime as an ideal in ...
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1answer
47 views

Commutative ring can be homomorphically mapped onto field

During my algebra lecture, my lecturer used the fact that any commutative ring can be homomorphically mapped onto a field. Is the statement true? How to show that? Thanks
2
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2answers
80 views

Why is $ \mathrm{Frac} ( A / \mathfrak{p} ) = A_{\mathfrak{p}} / \mathfrak{p} A_{\mathfrak{p}} $? [duplicate]

$ A $ is a commutative ring, $ \mathfrak{p} \in \mathrm{Spec} A $, $ A_{\mathfrak{p}} = ( A \backslash \mathfrak{p} )^{-1} A $, $ \mathrm{Frac} ( A / \mathfrak{p} )$ is the field of fractions of $ A / ...
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0answers
51 views

$M_{\mathfrak{p}} \otimes_{R_{\mathfrak{p}}} N_{\mathfrak{p}} = 0$ implies $M_{\mathfrak{p}} = 0$ or $N_{\mathfrak{p}} = 0$ [duplicate]

Studying commutative algebra I've found this statement: If $M$ and $N$ are finitely generated $R$-modules, with $R$ a commutative ring, and $\mathfrak{p} \subset R $ is a prime ideal, then ...
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1answer
42 views

Computing a regular sequence of generators for an ideal

Let $R = \mathbb{C}[x_1,\ldots,x_n]$. Let $I$ be an ideal, and suppose we know a finite list of generators for $I$, say $I = \langle f_1,\ldots,f_k\rangle$. Is this information enough to compute a ...
1
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0answers
42 views

Finitely generated idempotent ideal must be generated by an idempotent [duplicate]

Let $A$ be a commutative but not necessarily unital ring. How can we show that a finitely-generated ideal $I$ of a ring $A$ satisfying $I=I^2$ is generated by an idempotent element?
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1answer
43 views

Noether normalization and surjectivity (revisited)

Let $Y$ be an affine variety of dimension $d$ inside the affine space $\mathbb{A}^n$. Then $A(Y) = k[x_1,\dots,x_n]/I_Y=:k[\bar{x}_1,\dots,\bar{x}_n]$. By the Noether normalization theorem, there ...
5
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1answer
73 views

(Co)homology of free symmetric algebra

Let $V$ be a (co)chain complex, and let $Sym(V)$ be the free differential graded-commutative algebra generated by $V$. Definition and examples below in case you don't know what I mean. Question: ...
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4answers
73 views

Maximal ideal in the ring of polynomials over $\mathbb Z$

Let $\mathbb Z[x]$ the ring of polynomials with integers coefficients in one variable and $I =\langle 5,x^2 + 2\rangle$, how can I prove that $I$ is maximal ideal. I tried first see that $5$ and ...
0
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1answer
37 views

Relation between faithfully flatness and map of $Spec$

I'm stuck on this exercise ( from Bosch ) : Let $\phi :R \to R' $ a flat ring morphism. Show that $\phi$ is faithfully flat if and only if the associated map $Spec(R') \to Spec(R)$ , ...
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1answer
26 views

Canonical homomorphisms $R_{\mathfrak{p}_i} \to R/\mathfrak{p}_i^n$ are isomorphisms when $R$ is artinian

I'm doing this exercise (from the book of Bosch): Let $R$ be an Artinian ring and let $\mathfrak{p}_1, \ldots \mathfrak{p}_n $ be its (pairwise different) prime ideals. Show that: a) The ...
0
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1answer
42 views

Common equivalent conditions for Cohen-Macaulayness of a ring

I know the fact that a local ring $(R,m)$ with $\dim(R)=d$ is Cohen-Macaulay (C-M) if and only if any one of the following holds: 1) $\operatorname{grade}(m)=\operatorname{height}(m)$ 2) ...
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1answer
51 views

Rings which are finitely generated and free over Cohen-Macaulay rings are also Cohen-Macaulay

Let $S$ be a Cohen-Macaulay (C-M) ring, and $R$ a ring containing $S$ such that as an $S$-module is finitely generated free. Could we deduce that $R$ is also C-M? I guess probably we could use ...
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0answers
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A question about the nature of $D(f)$.

My book on Commutative Algebra says the following: Let $f$ be an element of commutative ring $A$. Then $D(f)=\text{spec }(A)/V(f)$ Say the ring under consideration is $\Bbb{Z_6}$. We know that ...
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1answer
151 views

Pseudo associated primes and short exact sequences

Let $A$ be a commutative ring, and $$0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow0$$ a short exact sequence of $A$-modules. The following inclusion relation is well-known: ...
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5answers
81 views

Show that $f^{-1}(\langle0\rangle)$ is not a maximal ideal of $\mathbb{Z}$.

Let $f\colon \mathbb{Z} \to \mathbb{Q}$ be a ring homomorphism. Show that $f^{-1}(\langle0\rangle)$ is not a maximal ideal of $\mathbb{Z}$.
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1answer
39 views

About second uniqueness primary decomposition theorem

I'm self-learning commutative algebra from Introduction To Commutative Algrebra of Atiyah and Macdonald and get frustrated about the second uniqueness primary decomposition theorem. I copy the theorem ...
0
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1answer
32 views

When a monomial ideal is primary

I know that a monomial ideal in $k[x_1, \ldots x_n]$ with $k$ a field is prime if and only if is of the following type $$I = (x_{i_1}, \ldots \ ,x_{i_k})$$ Is there a similar criterion to establish ...
0
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1answer
48 views

Height of finitely generated ideals in a catenary local ring

If $R$ is a noetherian local domain which is catenary, and $a_1,...,a_n$ are elements of the maximal ideal of $R$ with $\operatorname{height}(a_1,...,a_n)=n$, could we conclude that ...
1
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1answer
62 views

Primary decomposition of $I = (x^2, y^2, xy)$

I want to find a primary decomposition of the ideal $$ I = (x^2,y^2,xy) \subset k[x,y]$$ where $k$ is a field. How to proceed? Are there algorithms to find such decompositions? Where can I find ...
2
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1answer
88 views

Is the ring $ R = \{ f \in \mathbb{C}[x,y] \mid {\nabla f}(0,0) = (0,0) \} $ Noetherian?

Question: Is the ring $ R = \{ f \in \mathbb{C}[x,y] \mid {\nabla f}(0,0) = (0,0) \} $ Noetherian? I guess it isn’t Noetherian as I suspect that $$ (x y + y^{2}), \quad (x y + y^{2},x^{2} y + ...
1
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1answer
67 views

Finitely many prime ideals $\Rightarrow$ cartesian product of local rings

I'm stuck on this problem from Bosch, Algebraic geometry and commutative algebra: Let $R$ be a commutative ring containing only finitely many prime ideals and assume that a certain power of the ...
0
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1answer
44 views

Grade of non principal Prime ideals in Noetherian UFDs

I want to prove that in any Noetherian UFD the grade of every non-principal prime ideal is at least $2$. I say in a UFD $R$ each nonzero prime ideal contains a prime element. Since the given ...
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0answers
52 views

An example of regular local ring and its regular system of parameters

In the book of Eisenbud, "Commutative Algebra with a view toward Algebraic Geometry" it is quoted that "if $p$ is a prime integer, then $\mathbb Z_{(p)}[x_1,...,x_n]_{(p,x_1,...,x_n)}$ is a regular ...
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0answers
27 views

Criterion of separability of function on a curve.

Let $K$ be finitely generated extension of an algebraically closed field $k$ of transcendence degree $1$, $\operatorname{char }k =p$. Let $C(K)$ be set of discrete valuation rings $(\mathcal ...
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1answer
45 views

Affine $K$-algebra is Hilbert ring!? [duplicate]

We know that when $F$ is a field, the ring $F[x_1,...,x_n]$ is a Hilbert ring, because the field $F$ is a Hilbert ring. My questions: Is any (non-trivial) affine algebra over an algebraically ...
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1answer
39 views

Associated primes and Noetherian condition

Let $A$ be a commutative ring, $M$ an $A$-module, and $N\subset M$ a submodule. Consider the following two sets: $$\Omega:=\{\mathfrak{p}\in\operatorname{Spec}A \ | \ \mathfrak{p}=(N:m) \ \mbox{for ...
1
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1answer
55 views

Ring extension and Jacobson rings

If $R\subseteq S$ are commutative rings, is it a fact that $R$ is a Jacobson ring if and only if $S$ is so? I guess the contraction of maximal and prime ideals of $S$ may be helpful in this ...
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1answer
50 views

Dimension of a semilocal Hilbert ring is zero

Is the Krull dimension of any commutative semilocal Hilbert ring equal to zero? I appreciate any help from anyone!
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0answers
17 views

Verification of an argument regarding the multiplication of polynomials.

Let $A[[x]]$ be the ring of formal power series $\sum_{n=0}^\infty{a_nx^n}$, where $A$ is a commutative ring. Prove that if $f\in A[[x]]$ is a unit, then $a_0$ is a unit in $A$. Is the following ...
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1answer
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A question from Atiyah-Macdonald

Let $A$ be a commutative ring. I'm trying to prove that in $A[x]$, the Jacobson radical $\mathcal{J}$ is a subset of the nilradical $\mathcal{P}$. Let $a_0+a_1x+a_2x^2+\dots +a_nx^n\in\mathcal{J}$. ...
0
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1answer
43 views

Nullstellensatz non-valid for non-algebraically closed fields

I want an example (with details, please) showing that Nullstellensatz may be false over non-algebraically closed fields. Thanks in advance!
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2answers
59 views

intersection of non zero prime ideals of polynomial ring R[x] over integral domain R is zero

Let R be an integral domain. Then how to show that intersection of non zero prime ideals of R[x] is zero.
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3answers
121 views

Commutative ring with an ideal that contains all the nonunits

Is there an example of a commutative ring with an ideal that contains all the non-units? I was trying to think of some subring of $\mathbb Q$, but I couldn't get it to work.
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1answer
27 views

To show a certain integral closure is an order.

I have a Dedekind domain $R$ with field of fractions $K$ and a finite separable field extension $L$ of $K$. Let $S$ be the closure of $R$ in $L$. Is there a quick way to show that $S$ is finitely ...
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2answers
50 views

“Finitely generated as an $R$-module”

Please could somebody explain to me what it means for something to be finitely generated as an $R$-module? I can't seem to find a definition anywhere! Thanks!
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1answer
34 views

A local subring of $F[[x]]$?

Suppose that $F$ is a field and $R=F⊕x^2F[[x]]$, where $F[[x]]$ is the ring of power series in one indeterminate $x$ with coefficients in $F$. I guess that $R$ is a local ring with the maximal ...
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0answers
28 views

Relation between the initial ideal and radical

Let $I$ be an ideal of the polynomial ring $S$. Show that ${\rm In}(\sqrt I)\subseteq\sqrt{{\rm In}(I)}$, where by ${\rm In}(I)$ we denote the ideal of initial forms of I, In(I) = (In(f) : f $\in$ I). ...
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0answers
67 views

Infinitely many maximal ideals containing $x^2-y^3$

How can we find infinitely many maximal ideals containing the ideal $I = (x^2-y^3)$ of $\mathbb C[x,y]$?
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1answer
79 views

$\mathbb C[x_1,\ldots,x_n]/I=\mathbb C\times\cdots\times\mathbb C$.

Let $A=\mathbb C[x_1,\ldots,x_n]/I$ and for every $y\neq 0$, we have $y^2\neq 0$ and $\dim A=0$. I would like to prove that $A=\mathbb C\times\cdots\times\mathbb C$. Attempt of a solution $\dim ...
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1answer
40 views

Intersection of $max(R)$ with a closed subset in $Spec(R)$

Let $R$ be a commutative ring with unity and $E$ be a nonvoid closed subset of $Spec(R)$. If $U$ is an open subset of $Spec(R)$ with $E∩Max(R)⊆U$, where $Max(R)$ is the set of maximal ideals of $R$, ...
2
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1answer
60 views

Jacobson radical of a ring finitely generated over $\mathbb Z$

If a commutative ring $R$ with $1$ is finitely generated over $\mathbb Z$ could one deduce that the Jacobson radical of $R$ is nilpotent? I am aware of the well-known fact that when $R$ is ...
6
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0answers
98 views

If $a,b$ is an $R$-sequence, then $(ax-b)$ is prime [duplicate]

If $R$ is an integral domain, $a, b\in R, a\neq 0$ and $\bar b$ is not a zero divisor in $R/(a)$. I'm trying to prove $(ax-b)\in R[x]$ is prime. This question seems easy but I couldn't prove it, ...