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

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2
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1answer
71 views

When the injective hull is indecomposable

Let $R$ be a ring and $M$ an $R$-module. Denote by $E(M)$ the injective hull of $M$. I was trying to prove that the following conditions are equivalent: 1) $(0)$ is meet-irreducible in $M$; 2) ...
0
votes
0answers
118 views

1st uniqueness theorem of a primary decomposition

I have a problem with one step of the proof of the 1st uniqueness theorem of a primary decomposition in Atiyah, MacDonald Commutative Algebra. Theorem 4.5. Let $\mathfrak{a}$ be a decomposable ...
2
votes
2answers
177 views

Principal ideals having embedded components

Does there exist a noetherian domain $A$ and a principal ideal $I = (x)$ in it having an embedded component?
7
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2answers
147 views

What's the projective limit of these polynomial rings ?

Define an inverse system of polynomial rings over a commutative ring $k$ by the canonical projection $k[x_1,...,x_n] \to k[x_1,...,x_m]\;(m< n)$. Question: What is the projective limit ...
7
votes
1answer
177 views

Which definition of dimension came first?

In my algebraic geometry class, the dimension of an affine variety $X=V(I)$ was defined as the supremum of the length of chains of prime ideals in the coordinate ring $R=k[x_1,\ldots,x_n]/\sqrt{I}$, ...
4
votes
1answer
119 views

Why are projective modules contained in this class of modules?

Suppose $A$ noetherian and define $G(A):=\{M: M$ is an $A$-module reflexive and Ext$^i_A(M,A)=$Ext$^i_A(M^*,A)=0$ for $i\geq1\}$ Why are projective modules contained in this class? Of course if ...
5
votes
1answer
140 views

Family of prime ideals in $\mathbb{Z}[x,y]$

Problem: let $m$ be a positive integer. Find a necessary and sufficient condition on $m$ so that $I=(m, x^2+y^2)$ is a prime ideal in $R=\mathbb{Z}[x,y]$. An easy necessary condition is: $m$ is a ...
3
votes
1answer
77 views

An ideal is homogenous iff it is invariant under a certain automorphism.

I'm working on the following. Let $R=R_0+R_1+ \cdots $ be a graded ring and $u$ a unit of $R_0$. Then the map $T_u$ defined by $T_u(x_0+x_1+ \cdots +x_n) = x_0+x_1u+ \cdots + x_n u^n$ is an ...
1
vote
1answer
142 views

Proposition 2.14 iii) Atiyah Macdonald introduction to commutative algebra

Hi I am refering to proposition 2.14 in Atiyah-MacDonald introduction to commutative algebra and I cant find the bilinear maps that will induce A-module homomorphisms f,g where $f:(M \oplus N) ...
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0answers
36 views

Galois extension of a semilocal ring.

I wanna know if a finite Galois extension of a semi-local commutative ring is also semilocal.
17
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1answer
396 views

The polynomial whose roots are all real

Suppose $p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_0\in \mathbb{R}[x]$ is a polynomial whose roots are all real where $a_n=1$. We want to show that The polynomial $g(x)=\sum_{i=0}^{n} ...
3
votes
1answer
89 views

Concrete syzygy generators

Consider $g_1=x^2, g_2=y^2, g_3=xy+yz\in k[x,y,z]$ with a field $k$. We consider the reverse lexicographic order, and put $x>y>z$. I want to find the generators of the syzygies. Eisenbud CA ...
3
votes
1answer
220 views

Ideals of the tensor product $R\otimes_{k} S$?

Let $R$ and $S$ be commutative rings over a field $k$. Let $I$ be an ideal of the tensor ring $R\otimes_{k} S$. It is true that there exist ideals $I_{1}$ and $I_{2}$ of $R$ and $S$ respectively such ...
1
vote
0answers
45 views

Injectivity of maps on quotients of free but non-finite rank modules

Let $(A,\mathfrak{m})$ be a local noetherian domain. Suppose that $M$ and $N$ are free modules over $A$ and $f: M \rightarrow N$ is an $A$-module morphism. Suppose as well that $f$ is injective and ...
1
vote
1answer
126 views

Formal power series ring over a valuation ring of dimension $\geq 2$ is not integrally closed.

I recently tried exercise 10.4 in Matsumura's Commutative Ring Theory, but got stuck. The question is: If $R$ is a valuation ring of Krull dimension $\geq 2$, then the formal power series ring ...
7
votes
1answer
352 views

Homomorphism of local rings

Let $(A,\mathfrak{m})$ and $(B,\mathfrak{n})$ be local Noetherian rings. Suppose that $\phi : A\rightarrow B$ is a map such that $\phi(\mathfrak{m}) \subset \mathfrak{n}$ and suppose ...
2
votes
1answer
128 views

Equivalent definitions of associated prime ideals

I'm reading the book Undergraduate Commutative Algebra of M. Reid. He gives the following definition: Let $M$ be an $R$-module and let $P$ be an ideal of $R$. Then $P$ is an associated prime if: ...
5
votes
2answers
181 views

How to tell a ring is not discrete valuation Ring?

Suppose $K$ is a field and $R=K[x,y]/(y^2-x^3)$. The question requires to show the localization at $(x,y)$ is not a discrete valuation ring. I can find the unit element in the localization is the ...
3
votes
0answers
89 views

Applications of Govorov-Lazard Theorem?

The Govorov-Lazard Theorem states that a (right) module over an unital ring is flat iff it is a direct limit of finitely generated free (right) modules. I wonder if this theorem has interesting ...
6
votes
1answer
161 views

How to show a quasi-compact, Hausdorff space be totally disconnected?

This is from Atiyah-Macdonald. I was asked to show if every prime ideal of $A$ is maximal, then $A/R$ is absolutely flat, Spec($A$) is a $T_{1}$ space,further Spec($A$) is Hausdorff. The author then ...
3
votes
2answers
199 views

Finite length module and graded local duality

In the proof of Theorem 20.18 in Eisenbud Commutative Algebra, the following fact is stated: If $S=k[X_1,\ldots,X_r]$ and $N$ is a finite length graded $S$-module, then ...
6
votes
2answers
129 views

Is $\mathbb{C}[x]_{(x)}=\mathbb{C}[x]$?

I don't know what's the difference between $\mathbb{C}[x]_{(x)}$ and $\mathbb{C}[x]$. Isn't the localization is just equal to the original ring? Then why the first presentation is used?
2
votes
1answer
72 views

Question about principal prime ideals

Let $R$ be a commutative ring, and $I$ be a principal prime ideal. Is it true that $I$ does not contain any non-zero prime ideal? (You may assume that $R$ is Noetherian)
3
votes
2answers
112 views

What letter should I use to denote an ideal?

In commutative algebra, there seem to be two rather different notational conventions for ideals: either $I,J, \dots$ or $\mathfrak{a}, \mathfrak{b}, \dots$. By itself, it is hardly surprising - after ...
2
votes
1answer
144 views

Infinitely generated modules

Can you give me some examples of infinitely generated modules over commutative rings, other than $A[x_1,\ldots,x_n,\ldots]$? Thanks a lot!
3
votes
1answer
112 views

Question about prime ideals and union of ideals

The questions asks to show that if $A$ is a ring and $I, J_{1}, J_{2}$ ideals of $A$, and $P$ is a prime ideal, then $I \subset J_{1} \cup J_{2} \cup P$ implies $I \subset J_{1}$ or $J_{2}$ or $P$. ...
2
votes
3answers
193 views

Homomorphism of free modules $A^m\to A^n$

Let's $\varphi:A^m\to A^n$ is a homomorphism of free modules over commutative (associative and without zerodivisors) unital ring $A$. Is it true that $\ker\varphi\subset A^m$ is a free module? Thanks ...
2
votes
2answers
47 views

Determining a rule which assigns each ring a unit element

Probably this is easiest, but as I am somehow stuck I would be pleased about some comments. What I give myself is a rule $f$ which does the following: To every commutative ring $A$ with $1$ the rule ...
3
votes
1answer
58 views

Non trivial representation of zero in a module-finite extension

Suppose, $S$ is a ring, module-finite over a subring $R$. Let $s_1,...,s_n$ be a minimal set of generators of $S$ as an $R$-module. Can we have, $0=t_1s_1+...+t_ns_n$ for some $t_1,...,t_n\in R$, not ...
4
votes
1answer
183 views

Why are these two functors isomorphic?

Let $A$ be a local noetherian ring, $M$ an $A$-module finitely generated. Let $f$ be an $A$-regular and $M$-regular element (i.e. $f$ is not a zero divisors on $A$ nor on $M$). Then inside the ...
0
votes
2answers
175 views

Coproduct of two modules

Suppose that $M$ is an $A$-module, and $N$ is a $B$-module. The coproduct of $A$ and $B$ is $A\otimes_{\mathbb{Z}}B$, and the coproduct of $M$ and $N$ is $M\oplus N$. I was wondering if $M\oplus N$ ...
2
votes
1answer
199 views

Endomorphism algebra of the tensor product of modules

Let $k$ be a commutative ring, and let $M$ and $N$ be $k$-modules. Let $\mathrm{End}(M) = \mathrm{Hom}_k (M,M)$ be the endomorphism algebra. Is it true that $\mathrm{End}(M) \otimes \mathrm{End}(N) ...
1
vote
1answer
200 views

How to show $X=Spec(A)$ has maximal components $V(p)$?

This is from Atiyah Macdonald, Chapter 1, Execrise 20. If $A$ is a ring and $X=Spec(A)$, then the irreducible components of $X$ are the closed sets $V(p)$, where $p$ is a minimal prime ideal of $A$. ...
7
votes
1answer
380 views

Infinite coproduct of rings

I just learned from Wikipedia that coproduct of two (commutative) rings is given by tensor product over integers, and that coproduct of a family of rings is given by a "construction analogous to the ...
1
vote
2answers
103 views

What does it mean when elements act as units on a set?

I'm reading Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry, and I am confused by one of his proofs. The setup is that $R$ is a commutative ring, $U$ is a multiplicatively closed ...
2
votes
1answer
60 views

Some questions about rings

All rings are commutative and unital Q1: what means notation $$A\cong A_1\times\ldots\times A_n?$$ Is it true that elements of $A_1\times\ldots\times A_n$ are collection of elements of $A_1,\ldots ...
3
votes
2answers
268 views

Being maximal ideal follows from being a kernel

The ideal of all polynomials in $k[x_1,\ldots,x_n]$ with zero constant term is maximal (since it is the kernel of the homomorphism $k[x_1,\ldots,x_n]\to k$ which maps $f\mapsto f(0)$). I ...
2
votes
1answer
68 views

Behaviour of conductor ideal

Let $f : A \to B$ be a homomorphism of finitely generated $k$-algebras, where $k$ is a field. Let $J_A$ and $J_B$ denote the conductor ideals of $A$ and $B$ respectively for the corresponding ...
2
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0answers
92 views

Clarifications on Noether Normalization

I finished reading Noether Normalization but given that I have almost no prior algebra training I am concerned that my understanding is wrong. (Starting masters in Mathematics but previously was an ...
2
votes
0answers
225 views

Field of Fractions for Commutative Ring with Identity

I'm trying to complete a problem that is walking me through creating a field of fractions $F$ from a commutative ring with identity (and no non-zero divisors) $R$. The construction first asks me to ...
1
vote
1answer
362 views

Modules of finite type over Noetherian rings

Let M be a unitary module of finite type over a commutative Noetherian ring R with a unit. Can M then always be represented as a quotient of a pair of free R-modules of finite-type?
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votes
2answers
65 views

Elementary Question about Torsion Subgroups

Let $G$ be an abelian group which is killed by multiplication with the integer $n\geq 1$. Let $n=a\cdot b$ with $a,b \geq 1$ and relatively prime. Denote by $G[a]$ resp. $G[b]$ the $a$-resp. ...
1
vote
1answer
81 views

Relation between DVR's of a local domain and localizations of its integral closure.

$\textbf{1.}\,\,\,\,\,\,\,\,$ Let $(A,\mathfrak m_A)$ be a one dimensional local domain and let $B$ be its integral closure in the fraction field $L=\textrm{Frac}\,A$. Assume that $B$ is finitely ...
6
votes
1answer
207 views

polynomials over a local Artinian (or finite) ring

In this question " Zero-divisors and units in $\mathbb Z_4[x]$ " it looks like it has been shown that the set of zero divisors of $\mathbb{Z}_4[x]$ coincides with its nilpotent elements. Since the ...
1
vote
1answer
68 views

Some question on filtrations

Let $S$ be a noetherian ring and $M$ a finitely generated $S$-module. There exists a filtration by submodules $$0=M_0 \subseteq M_1\subseteq \cdots \subseteq M_r=M.$$ I want to show that for any ...
2
votes
0answers
341 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
88 views

Flat module over A implies flat module over B

Let $A,B$ be commutative rings with 1, $\varphi:B\rightarrow A$ a ring homomorphism and M an $A$-module. If M is flat as an $A$-module, is it also flat as a $B$-module? (The structure of $B$-module is ...
3
votes
1answer
423 views

Tensor product of projective modules

(All rings are commutative) Let $A$ be a noetherian ring. Let $B$ be a noetherian $A$-algebra (not nessecerily f.g!) Suppose $M$ and $N$ are finitely generated projective $B$-modules (for my ...
4
votes
2answers
181 views

Galois theorem for ideals?

Let $R$ be an integrally closed integral domain with fraction field $K$. Let $L$ be a finite Galois extension of $K$ and let $\sigma_1,\dots,\sigma_n$ be the elements of $Gal(L/K)$. Let $S$ be $R$'s ...
4
votes
2answers
155 views

When does $\mathfrak{a}B\cap A = \mathfrak{a}$?

Let $A\subset B$ be rings, and let $\mathfrak{a}$ be an ideal of $A$. Under what circumstances does $\mathfrak{a}B\cap A = \mathfrak{a}$? More precisely, are there conditions on $A,B$ that guarantee ...