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

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3
votes
4answers
335 views

Proof that the ideal $(xy, xz)$ in $\mathbb{A}^3$ is radical but not prime

The proof that $(xy, xz)$ is not prime seems easy. In particular, $xy \in (xy, xz)$, but neither $x$ nor $y$ is in $(xy, xz)$. On the other hand, I don't know how to prove that $(xy, xz)$ is ...
0
votes
1answer
31 views

extending an integral domain by an integral element

Let $A$ be an integrally closed integral domain, let $K$ be its field of fractions and $\bar{K}$ the algebraic closure of $K$. Let $t \in \bar{K}$ be integral over $A$. By a known theorem, the minimal ...
3
votes
1answer
50 views

does tensoring commute with taking images without flatness?

This is a follow-up to this question: was flatness really used in this argument? (Matsumura, Theorem 7.2). The author of the given answer says we need flatness to ensure that tensoring commutes with ...
2
votes
1answer
73 views

was flatness really used in this argument? (Matsumura, Theorem 7.2)

Let $A$ be a ring and $M$ an $A$-module. Then $M$ is faithfully flat over $A$ $\Leftrightarrow$ $M$ is flat over $A$ and $M \otimes N=0 \Rightarrow N=0$. This is part of theorem 7.2, p. 47 in ...
1
vote
1answer
88 views

proving closure of a set of automorphisms in the Krull topology in the context of integral ring extensions

Let $A$ be an integrally closed integral domain with field of fractions $K$ and let $p \in Spec(A)$. Let $L/K$ be a Galois extension with group $G$ and $L'/K$ be a finite Galois subextension. Let $B$ ...
-2
votes
1answer
113 views

Spec($\mathbb Q \times \mathbb Q$)

What is Spec($\mathbb Q \times \mathbb Q$)? Where Spec(R) denotes the set of all prime ideals of R
3
votes
1answer
149 views

Generic point of a curve in affine plane

Consider the affine plane over $k$, i.e. Spec $k[x,y]$. There are three kinds of prime ideals: $(0)$, $(x-a,y-b)$, and $(f(x,y))$, for $f$ irreducible. Let the ideal $(f(x,y))$ correspond to the point ...
2
votes
1answer
115 views

increasing union of finitely generated submodules of M need not be finitely generated

Show by an example that an increasing union of finitely generated submodules of M need not be finitely generated. I was thinking about $R[x_1,x_2,x_3,....]$. Then if we consider the ideal ...
1
vote
3answers
660 views

$R$ has just one prime ideal iff $R/N(R)$ is a field

Let $R$ be a ring with nilradical $N(R)$. How can one show that $R$ has just one prime ideal if and only if $R/N(R)$ is a field ?
0
votes
1answer
48 views

A bound on the codimension of a ring

Assume $R$ is a local ring and that $R=S/I$ with $S$ regular. How can I prove that the minimum number of generators of $I$ is greater or equal to $\dim S-\dim R$ and that equality holds if and only if ...
0
votes
1answer
38 views

A bound on the first deviation of a local ring

Assume $R$ is a local ring and $R=Q/I$ with $Q$ regular. Let $\varepsilon_1(R)$ be its first deviation (the dimension as a vectorspace of the first Koszul homology). How can I prove that the first ...
1
vote
1answer
47 views

If $\mu_r(m)=rm$ then ker $(\mu_r)$ and coker $(\mu_r)$ are modules over $R/I$

Let R be a commutative ring with unity and M be an R-module. Let $\mu_r:M\rightarrow M$ be the map $\mu_r(m)=rm$. Then prove that ker $(\mu_r)$ and coker $(\mu_r)$ are both modules over $R/I$ where ...
2
votes
2answers
258 views

Factoring 1001 in $\Bbb Z[\sqrt 7]$

I am solving the problem of factoring 1001 into prime elements in $\Bbb Z[\sqrt 7]$. I have a couple of questions regarding this. It seems that $\Bbb Z[\sqrt 7]$ is an Euclidean domain. But I do ...
0
votes
3answers
135 views

Noetherian ring and ascending chain of ideals.

Let $A$ be a noetherian ring, $\mathfrak a_{0}$ an ideal in $A$ and $$S=\{ n\in \mathbb{N}\mid \text{there exist ideals }(\mathfrak a_{i})_{i=1,\dots,n}\text{ such that }\mathfrak a_{0}\subset ...
3
votes
1answer
60 views

homomorphism of graded modules

This is a little exercise problem from Peeva's book on graded syzygies. Let $\phi:N\to T$ be a homomorphism of graded $R$-modules. If $f=f_1+\cdots+f_n\in N$ and $f_i$ are its homogeneous components, ...
2
votes
2answers
205 views

Question about Matsumura book Commutative Ring Theory.

I'm reading the book of Matsumura, Commutative Ring Theory and I can't understand a statement that he does, namely: If $A$ is ring, $M$ a finite $A$-module, $\mathfrak p\in \mathrm{Spec}(A)$, ...
0
votes
2answers
243 views

Show the ideal $I=(x^{2}-y,z-1)$ is prime in $K[x,y,z]$

I am tasked to show that the ideal $I=(x^{2}-y,z-1)\subset K[x,y,z]$ is it's own radical where $K$ is an algebraically closed field. I tried to proceed in the obvious fashion. Let ...
2
votes
2answers
351 views

Strategy to prove that rings are UFD's

I was looking for strategies to prove when rings are going or not to be UFD's. I really only know that if I manage to prove that there is an element on the fraction field $K$ of my ring $R$ that is ...
0
votes
1answer
78 views

conjugate prime ideals of integral extensions and relevance of the characteristic of the ground field

This question refers to the proof of theorem 9.3, p. 66 in Matsumura's Commutative Ring Theory: "if $A$ is an integrally closed domain, $K$ its field of fractions and $L/K$ a normal field extension, ...
2
votes
0answers
69 views

Stable fiber products of commutative rings

Let $R_1 \to T$ and $R_2 \to T$ be homomorphisms of commutative rings. Consider the fiber product $R=R_1 \times_T R_2$. Let $R \to R'$ be a homomorphism of commutative rings, and define $R'_i$ to be ...
3
votes
1answer
162 views

Is $R/I$ flat over $R$ when $I$ is the nilradical of $R$ and $I=I^2$?

This is a flowup of this question. To be precise, let $R$ is a commutative unitary ring with nilradical $I$. In the above URL, it is proved that if $R/I$ is a flat $R$-module, then $I=I^2$. My ...
1
vote
1answer
160 views

$\mathrm{Tor}_1(R/a,M)$ and $\mathrm{Ext}^1_R(R/a,M)$, $a\in R$ a non-zero divisor

In Lecture Notes in Algebraic Topology, Davis & Kirk, it is written: Proposition $\mathbf{2.4.}\,\,$ Let $R$ be a commutative ring and $a\in R$ a non-zero divisor (i.e. $ab=0$ implies $b=0$). ...
2
votes
0answers
51 views

Some property of an ideal in commutative ring [duplicate]

Let $R$ be a Dedekind domain, and let $\mathfrak{m}$ be a maximal ideal in $R[x]$ is of the form $\mathfrak{m} = (\mathfrak{p},f(x))$ where $\mathfrak{p}$ is a maximal ideal in $R$, and $f$ is a ...
7
votes
1answer
143 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 subset of ...
2
votes
1answer
125 views

Elementary method for finding $I(Y)$ for the curve $Y$ defined parametrically by $x=t^{3}$, $y=t^{4}$, $z=t^{5}$

In order to motivate some of the theory we will be learning in a computational commutative algebra course, my professor assigned a number of computational problems that are [seemingly] quite difficult ...
6
votes
1answer
139 views

Localness of the UFD Property

If $A$ is a noetherian domain and $A_p$ is a UFD for some prime ideal, is there some $f$ not contained in $p$ such that $A_f$ is a UFD?
6
votes
1answer
397 views

Primary decomposition of an ideal (exercise in Reid) [duplicate]

I would like to understand how to use geometry to solve a problem from Reid's book on commutative algebra. The problem is the following Let $k$ be a field and consider the ideal $I = (xy, x - yz) ...
3
votes
2answers
157 views

Universal Property of the Exterior Algebra

Let $k$ be a field and let $A$ be a commutative algebra over $k$. I want to calculate the exterior algebra $\Lambda_A^\bullet A$. We have $\Lambda_A^0 A = \Lambda_A^1 A= A$, and $\Lambda_A^k A = 0$ ...
6
votes
2answers
254 views

A (probably) wrong exercise from Morandi's Field and Galois theory

After some efforts I realize that the following exercise is wrong: (rings are unitary throughout the book) Morandi's Field and Galois Theory, Appendix A, exercise 18 (b) Let $A\subseteq B$ ...
1
vote
1answer
86 views

On verifying Proj S is a scheme

In Hartshorne II Prop 2.5, it says $D_+{(f)}$ is homeomorphic to $\text{Spec}(S_{(f)})$, but I cannot prove it. Since $D_+{(f)}$ homeomorphic to $S_f$, I have to show $\text{Spec}(S_{(f)})$ ...
3
votes
1answer
383 views

Quotient ring is an UFD

Prove that the ring $\mathbb R[x,y,z]/(x^2+y^2+z^2-1)$ is an unique factorization domain.
2
votes
0answers
87 views

Two problems in commutative algebra

Let $A$ be a local ring and $L$ and $F$ be free $A$-modules of finite rank. Consider bases $e_1,\dots,e_m$ and $f_1,\dots,f_n$ of $L$ and $F$. To a homomorphism $u:L\rightarrow F$ associate the matrix ...
12
votes
1answer
259 views

Coprime elements in finite rings

Let $R$ be a finite commutative ring. Consider elements $a,b \in R$ such that $Ra+Rb=R$. A paper I'm reading asserts that there exists some $x,y \in R$ such that $x(a+yb) = 1$. Of course, it ...
4
votes
1answer
119 views

PID modulo a non-zero ideal is a semilocal ring

Let $R$ be a commutative ring, $\mathfrak{m}\subset R$ a maximal ideal and $f$ a monic polynomial in $R[x]$. I want to show that $A:=\frac{R[x]}{\mathfrak{m}[x]+(f)}$ is a semilocal ring, where ...
1
vote
2answers
144 views

How to compute $I(Y)$ for the curve $Y$ defined parametrically by $x=t^{3}$, $y=t^{4}$, $z=t^{5}$?

Let $Y\subset\mathbb{A}^{3}$ defined parametrically by $x=t^{3}$, $y=t^{4}$, and $z=t^{5}$. I want to compute $I(Y)$. I think that $$I(Y)=(x^{20}-z^{12},x^{20}-y^{15},y^{15}-z^{12}),$$ but I ...
0
votes
2answers
92 views

Prove $(x^3, xy)$ = $(x)\cap(x^3,y)$

Prove $(x^3, xy)$ = $(x)\cap(x^3,y)$ in $K[x,y]$. In a similar way find an irreducible decomposition for $I=(x^3, xy, y^2)$. Here I tried in this way as: Let $r \in (x^3, xy) $ then $r= ax^3+b ...
4
votes
1answer
109 views

$\langle x^a, y^b\rangle $ is an irreducible ideal in $K[x,y]$

Prove that $\langle x^a, y^b\rangle$ is an irreducible ideal in $K[x,y]$. Any kind of help is very much welcomed.
1
vote
0answers
92 views

How to find an invertible maximal minor of a matrix

Let $A$ be a local ring and $L$ and $F$ be free $A$-modules of finite rank. Consider bases $e_1,\dots,e_m$ and $f_1,\dots,f_n$ of $L$ and $F$. To a homomorphism $u:L\rightarrow F$ associate the ...
7
votes
2answers
184 views

Birational and faithfully flat $\implies$ isomorphism

Let $A \subseteq B$ be integral domains with the same field of fractions. Assume that $A \to B$ is faithfully flat. Why do we have $A=B$? This is an exercise in Matsumura's book. Here is my idea: If ...
5
votes
2answers
297 views

Vandermonde matrices over a commutative ring.

Suppose that $R$ is a commutative ring with identity. I am trying to prove that the two following statements are equivalent. The ideal generated by all determinants of $n\times n$ Vandermonde ...
6
votes
1answer
193 views

A mistake in the proof of Lazard's theorem in Eisenbud's book on Commutative Algebra?

In Prof. Eisenbud's book Commutative Algebra with a View Toward Algebraic Geometry, there is in Appendix 6.2 a proof of Govorov & Lazard theorem that seems to me slightly wrong. It is written ...
4
votes
1answer
232 views

Minimal number of generators of an ideal in $\mathbb{C}[x,y,z]$

In the polynomial ring $\mathbb{C}[x,y,z]$, I should prove that the ideal $$I=(\underbrace{x^4-y^3}_{=:\,p_1},\underbrace{x^5-z^3}_{=:\,p_2},\underbrace{y^5-z^4}_{=:\,p_3})$$ can not be ...
2
votes
1answer
112 views

Meaning of the Rank of a Map of Free Modules?

I am reading the section on differentials in Eisenbud's book (Commutative Algebra), and I'm just wondering what he means in sentences like this one: "Suppose that $J:R^t \rightarrow R^r$ is a map of ...
11
votes
1answer
169 views

Polynomials invariant under the action of $S_m \times S_n$

The polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ has a maximal subring invariant under the action of $S_n$ on the variables. This is the ring of symmetric polynomials. Suppose we have ...
2
votes
0answers
180 views

Maximal ideal in commutative ring

Let $R$ be a Dedekind domain, $\mathfrak{m}$ be a maximal ideal in $R[x]$ is of the form $\mathfrak{m} = (\mathfrak{p},f(x))$ where $\mathfrak{p}$ is a maximal ideal in $R$, and $f$ is a polynomial in ...
1
vote
1answer
117 views

A reflexive module which is not free

Here is an exercise in Christian Peskine's book An Algebraic Introduction to Complex Projective Geometry, pg. 25: Show that $(X_0,X_1)/(X_0X_3-X_1X_2)$ is a reflexive but not free ideal of the ...
6
votes
1answer
135 views

Is the $\mathfrak m$-adic completion of a radical ideal again a radical ideal?

Let $(R,\mathfrak m)$ be a local (Noetherian) ring and let $\hat{R}$ be its $\mathfrak m$-adic completion. Let $I$ be an ideal of $R$ which is a radical ideal. Is it then also true that $\hat{I}$ ...
0
votes
1answer
36 views

Whether an isomorphism fixes the elements of the ground field

Suppose $k$ is an algebraically closed field and $A:=k[x_1,\ldots ,x_n]$. Now, let $M$ be a maximal ideal of $A$ and suppose $f:A/M\to k$ is an isomorphism. Then, is it true that $f(a+M)=a$ for all ...
2
votes
1answer
81 views

A dimension problem in commutative algebra

Let $A$ be a commutative ring ($A$ has an identity element), $\mathcal{I}$ is a non-zero ideal of $A$. If $\mathcal{I}$ is a free $A$-module, then we can suppose $\mathit e_{1}, \mathit ...
3
votes
1answer
258 views

When is the generic point of an integral noetherian scheme open (reference)?

Let $X$ be an integral noetherian scheme, let $\xi$ be its generic point. Then it is not so hard to show that $\{ \xi\}$ is open in $X$ if and only if $X$ is a finite set. In termes of algebra, it ...