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

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Associated primes of $M/IM$

Let $R$ be a Noetherian ring, $I$ an ideal of $R$ and $M$ an $R$-module. Is the following formula true? $$Ass(M/IM)=Ass(M)∩V(I)$$ Thanks.
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0answers
26 views

Question of a proposition about direct product

I try to prove it's injective, surjective and homomorphism. define f(x)=(x+a1,x+a2,....,x+an),it's homomorphism. it's injective <=> the intersection of ai=0 I don't know how to prove the ...
1
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2answers
91 views

Why is $\mathbb{Q}[X,Y]/(X^2+Y^2-1)$ a Dedekind domain?

What is the best way to understand that $D:=\mathbb{Q}[X,Y]/(X^2+Y^2-1)$ is a Dedekind domain? I first noticed that $X^2+Y^2-1$ is irreducible in $\mathbb{Q}[X,Y]$ since it is $Y-1$ Eisenstein in ...
0
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1answer
45 views

Is this automorphism the identity map

Let $A$ be a commutative ring and let $f: A \rightarrow A$ an surjective homomorphism, let $a$ be a ideal of $A$ then if $f(a)\subseteq a$ then it's $f$ is the identity map, or not necessary.
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2answers
41 views

Quotient ring is cyclic group implies every ideal is generated by 2 elements

I'm trying to solve the following exercise: Let $R$ be a commutative ring with identity. If for every ideal $\mathfrak{a} \neq 0$ of $R$ we have ($R/\mathfrak{a}$,+) is a cyclic group then ...
2
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1answer
37 views

If an identity in the language of rings holds for all fields, does it necessarily hold for all commutative rings?

It is weirdly difficult to find new identities for ring theory (other than commutativity) that make it more like field theory. This motivates my: Question. If an identity in the language of rings ...
2
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1answer
42 views

Dimensions of integral ring extensions

If $X$ is a commutative ring with identity and $Y$ is an integral $X$-algebra, show that $\dim\,X=\dim\,Y$. I think also that $X$ needs to be a subring of $Y$. Why is this true?
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2answers
74 views

Tensor product of quotient rings [duplicate]

$A$ is a commutative ring with unit and $\mathfrak a$, $\mathfrak b$ ideals. I have to show that $$A/\mathfrak a \otimes_{A} A/\mathfrak b \cong A/(\mathfrak{a+b}).$$ Any hint ?
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1answer
32 views

With B integral over subring A, homomorphism from A to algebraically closed field F can be extended to B.

Here's the problem I am working on: Let A be a subring of B such that B is integral over A, and let $f: A \rightarrow F$ be a homomorphism of A into an algebraically closed field F. Show that f ...
2
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1answer
45 views

$A_{p}$ is a field when $p$ is a minimal prime and $A$ reduced

$A$ is a reduced commutative ring with unit; $p$ is a minimal prime ideal. If $S = A \setminus{p}$ , I have to show that the ring $A_{p} = S^{-1}A$ is a field. My thoughts: Since $p$ is a minimal ...
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1answer
47 views

proposition 1.10 ii) A&M Introduction of commutative algebra

I am working through Introduction of commutative algebra and am having trouble with the following question: (I'll use f instead of the map,since I don't know how to input it.) Q1: Why there exist ...
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1answer
28 views

is there a counterexample of this map isn't surjective?

The ring A is a commutative ring with identity. I think ii) is true if they are not coprime. because for every (x+a1,....,x+an) we can find a x such that f(x)= (x+a1,....,x+an). Could you please ...
0
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1answer
27 views

Maximal multiplicative set and minimal prime ideal theorem proof [duplicate]

Let A be a ring and P a prime ideal included in A. Show that A∖P is a maximal multiplicative set if and only if P is a minimal prime ideal of A. What can be the proof for this theorem ?
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0answers
34 views

Sum of three squares of polynomials

In the polynomial ring $\mathbb{R}[x,y,z,u,v,w]$ with $6$ variables, I would like to know if there exist $f,g,h\in \mathbb{R}[x,y,z,u,v,w]$ such that $$ (x^2+y^2+z^2)(u^2+v^2+w^2)=f^2+g^2+h^2. $$ Any ...
1
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0answers
72 views

Calculate the primary decomposition

Consider the polynomial ring $R=K[x_1,\ldots, x_8]$ over field $K$. Set $\mathfrak{p}_1=(x_1, x_2, x_5, x_6)$, $\mathfrak{p}_2=(x_3, x_4, x_7, x_8)$ and $I=\mathfrak{p}_1\cap \mathfrak{p}_2$, ...
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1answer
31 views

Simple integral extension question

If $R$ is a commutative ring, why is every $x$ in $R$ integral over $R$? I can't see what monic polynomial will have $x$ as a root.
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1answer
65 views

Maximal multiplicative set and minimal prime ideal

Let $A$ be a ring and $P$ a prime ideal included in $A$. Show that $A \setminus P$ is a maximal multiplicative set if and only if $P$ is a minimal prime ideal of $A$. What can be the proof for this ...
0
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1answer
56 views

Invertible elements and maximal ideals of a localization

Let $n\in\mathbb Z$ and let $A$ be the set of integers co-prime to $n$. Denote $A^{-1}\mathbb Z$ by $\mathbb Z_{(n)}$. 1) Find the invertible elements of $\mathbb Z_{(6)}$ My attempt: let $m$ be ...
2
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1answer
45 views

Prime Spectrum of A Ring

I was given the definition that the spectrum of a ring R, denoted Spec R, is the set of the prime ideals of R. Then for an arbitrary subset $S \subseteq R$, then $V(S) = \{P \in SpecR | S \subseteq R ...
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1answer
131 views

Characterizing the field of fractions of $\mathbb Q[x,y]/(x^2+y^2-1)$.

Let $A = \mathbb Q [x, y] / (x ^ 2 + y ^ 2 - 1)$ and note that $A$ is a domain. How to show that $\operatorname{Quot} (A)$ (or $\operatorname{Frac} (A)$, i.e. the "field of fractions") is isomorphic ...
2
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1answer
35 views

Example of $\sum_i a_i\otimes b_i\in M\otimes_AN$ which cannot be written as $a\otimes b$

In the appendix of my commutative algebra text: Note that in general the element of $M\otimes_AN$ is a sum of the form $\sum_i a_i\otimes b_i$ and cannot be necessarily written as $a\otimes b$. ...
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1answer
46 views

Quotient field of a localization

I have a basic question about rings of fractions. Let $R$ be a commutative integral domain with quotient field $K$, $\mathfrak p$ a non-zero prime ideal of $R$ and $R_{\mathfrak p}$ the localization ...
2
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1answer
149 views

Intuition? how the author reaches the answer?

I've a question on 2 problems in this book: 2.4. Let $S = K[x_1, . . . , x_6]$. Let $f = x_1x_5 − x_2x_4$, $g = x_1x_6 − x_3x_4$ and $h = x_2x_6 − x_3x_5$. (a) Find a monomial order $<$ ...
2
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0answers
46 views

Example of irreducible ideal which is not strongly irreducible

I have read a paper with title Ideal Theory in Commutative Semirings by Reza Ebrahimi Atani and Shahabaddin Ebrahimi Atani. In this paper we have the following definitions: An ideal I is irreducible ...
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1answer
46 views

How to find a chain of prime ideals in $\mathbb{Z}[x]$ [duplicate]

How can I build three prime ideals of $\mathbb{Z}[x]$, $P_1, P_2, P_3$ with $P_1 \subsetneq P_2 \subsetneq P_3$ and justify this?
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1answer
55 views

Prime ideals of Z[x]

how to build three prime ideals of Z [x] (P_1, P_2, P_3) as P_1 is strictly included in P_2 and P_2 and strictly included in P_3?
3
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1answer
36 views

$\operatorname{rank}(F) = \operatorname{dim}_{k}(\frac{F}{mF})$

Let $R$ be a commutative ring with unit; $m$ is a maximal ideal; $F$ a free $R$-module. We know that $\frac{F}{mF}$ is a vector space over $\frac{R}{m} = k$ . I have to prove that ...
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0answers
40 views

Cohen-Macaulay over a tensor product of rings

Let $A, B, C$ be Noetherian Rings such that $A$ is a subring of $B$ and there exists a ring homomorphism $A \rightarrow C$. Let $M$ be a $(B,C)$ bimodule, i.e. $M$ is both a $B$-module and a ...
2
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2answers
54 views

“Primeness” of C[x] in B[x], where A is a subring of B and C is the integral closure of A in B.

Let A be a subring of B, and C the integral closure of A in B. If f, g are monic polynomials in B[x] such that fg is in C[x], then f, g are in C[x]. The first part of the problem allowed the ...
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0answers
32 views

Ring A is integral over the subring of invariants under a finite group action

I need to prove that if G is a finite group that acts on ring A, and $A^G$ is the subring consisting of elements of A which are invariant under all g in G, then A is integral over $A^G$. The hint ...
4
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1answer
155 views

showing Cohen-Macaulay property is preserved under a ring extension

Let $R$ be an $\mathbb{N}$-graded Noetherian ring, with $R_0$ local Artinian. Assume also that $R$ is finitely generated over $R_0$ by elements of degree $1$. Let $M$ be a Cohen-Macaulay $R$-module. ...
3
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0answers
33 views

On the Bass numbers of a local ring

Assume $R=k[x,y]/(x^2,xy,y^2)$, I would like to calculate the dimension as $k$-vectorspace of $\mathrm{Ext}^i_R(k,R)$. I see that as vector-space $\mathrm{Ext}^i_R(k,k)\cong k^{2^{i+1}}$, is it true ...
2
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1answer
47 views

A gentle reference for flat modules with exercises

I would like to learn about sources that give a gentle introduction to flat modules, with exercises. Could you recommend me some source? This might be a section in a book, or some article. I am ...
2
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0answers
48 views

Normal ring and unmixed ideals

Let $R$ be a commutative Gorenstein local ring , $I$ an ideal of $R$ . If $R/I$ is normal ring , then for any $p \in \operatorname{Ass_{R}}(R/I)$, $\operatorname{ht}(p)= \operatorname{ht}(I)$?
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2answers
68 views

If $A/\mathfrak a$ is flat over $A$ then $V(\mathfrak a)$ is open. Why?

I am trying to understand the following statement. Let $A$ be a noetherian commutative ring and $\mathfrak a\subset A$ is an ideal. Suppose that the ring $A/\mathfrak a$ is flat over $A$, then ...
2
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1answer
43 views

Relation Between Free Quotients and Modules

Here is my question: Let $M$ and $M'$ be $R$-modules, where $R$ is a commutative ring, and $N \subseteq M$ and $N' \subseteq M'$ submodules. Suppose that $N \cong N'$ and $M/N \cong M'/N'$. Determine ...
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0answers
33 views

Generators of a monomial Ideal

I am trying to determine the set that generates a monomial ideal. Namely, the ideal $(xy,yz,xz)^3$. I know it will have terms $x^3y^3$, $z^3y^3$, $x^3z^3$. For the other terms that generate, I do ...
4
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1answer
71 views

Don't understand a proposition and its proof

Proposition 5.1 from Commutative Algebra by Atiyah and Macdonald: $x∈B$ is integral over $A$,then $A[x]$ is a finitely generated $A$-module. The elements in $A[x]$ are the set of all the sum. If ...
1
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1answer
25 views

Help to understand Noether Normalization Theorem

I'm trying to understand the statement of the Noether Normalization Theorem: How can $k[x_1,\ldots,x_n]$ be equal to $k[x]$? a typical element of $k[x]$ is for example $ax^2+bx$ with $a,b\in k$, ...
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2answers
432 views

How does Hilbert's Nullstellensatz generalize the “fundamental theorem of algebra”?

What is Hilbert's Nullstellensatz in the sense of the generalization of "fundamental theorem of algebra"? I've seen that in some texts it was referred to as the generalization of the fundamental ...
1
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1answer
132 views

Support of $\operatorname{Hom}(R/I, M)$

Let $R$ be a Noetherian ring, $I$ be an ideal of $R$ and $M$ be an $R$-module. Is the following formula true? $\operatorname{Supp}\operatorname{Hom}(R/I, M)=\operatorname{Supp}(M) \cap V(I)$ If ...
2
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1answer
63 views

Minimal primes and zero divisors

Let $R$ be a commutative local ring, $M$ a finitely generated $R$-module, and $x \in M$. Is it true that if for any $p \in$ $\operatorname{Min}(R)$ there exists $a_{p}\notin{p}$ such that $a_{p}x=0$, ...
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2answers
34 views

Spec($A$) is connected if $A$ is local

Another exercise from Balwant-Singh: Show that if $A$ is local then Spec($A$) is connected in the Zariski topology. Any hint ?
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1answer
42 views

Idempotent/Spec

I'm studying Basic Commutative Algebra by Balwant-Singh; I'm stuck on this exercise: $A$ is a commutative ring; show this $3$ conditions are equivalent: 1) $A$ contains a non-trivial idempotent 2) ...
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1answer
53 views

Is it true that an ideal is primary iff its radical is prime?

Is it true that an ideal $I$ in a commutative ring is primary iff $Rad(I)$ is prime? If not, what are some nice counterexamples?
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53 views

A multiple choice question on noetherian and artinian modules

Let $R$ be a commutative ring with identity, $M$ a finitely generated $R$-module and $A$ an Artinian $R$-module. Which of the following statements is true? (1) $\mathrm{Hom}_{R}(A,A)$ is a finite ...
14
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2answers
140 views

If $M\otimes N=R^n$ need $M$ be projective?

So if over a commutive ring, $R$, we have that $M\otimes N=R^n$, $n\neq 0$, need we have that $M$ and $N$ be finitely generated projective? We have finite generation, because if $M\otimes N$ is ...
3
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1answer
38 views

Showing the set of diagonalizable matrices is constructible

Identifying $M_n(k)$ with $k^{n^2}$ with $k$ algebraically closed, I am asked to show that the subset of diagonalizable matrices, $D_n$ is constructible. Constructible is defined as being the finite ...
4
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1answer
89 views

is the hilbert polynomial integer-valued everywhere?

Let $R$ be an $\mathbb{N}$-graded Noetherian ring, finitely generated over $R_0$ with $R_0$ local Artinian. Let $M$ be a finite $R$-module of Krull dimension $d$. It is known that the Hilbert function ...
2
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0answers
52 views

A certain valuation of $k(X,Y)$ with value group $\mathbb{Z}+\mathbb{Z}\alpha$

Let $k$ be a field, $X$ and $Y$ indeterminate, and suppose that $\alpha$ is a positive irrational number. Then the map $\nu:k[X,Y]\rightarrow \mathbb{R}\cup \{\infty\}$ defined by taking $\sum ...