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

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is there a criterion that says whether an ideal is radical or not?

Let $R=k[x,y,z]$. Is there a criterion that says whether an ideal of $R$ is radical or not? thanks
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50 views

About a class of commutative rings that they have maximal ideals for any element non-inversible in $ZF\neg AC $

Let $\mathcal{N}{oetherian}\mathcal{C}\mathcal{R}{ng} \overset{def}{=} {\left\lbrace{ R \in \mathcal{C}\mathcal{R}{ng} \wedge R \,\text{is}\, \mathcal{N}{oetherian} }\right\rbrace}$. I define the ...
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78 views

Too many independent cubic polynomials in an ideal $I\subset \mathbb C[x,y,z]$

Let us consider the ideal $I=(x^2-x,y,xz)\subset \mathbb C[x,y,z]$. I want to prove that $I$ contains (exactly) $5$ linearly independent polynomials of degree $3$. In three variables, we have ...
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1answer
99 views

$k[x]/(x^n) \otimes_{k[x]} k[y]/(y^m)$

This is part of an exercise from Eisenbud: $k$ is a field, describe as explicitly as possible a) $k[x]/(x^n) \otimes_{k[x]} k[y]/(y^m)$ b) $k[x] \otimes_{k} k[y]$ Any hint ?
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1answer
114 views

What are the maximal ideals of $\mathbb{Z}[t,t^{-1}]\otimes \mathbb{Q}$?

I know that $\mathbb{Z}[t,t^{-1}]$ is a localization of $\mathbb{Z}[t]$, the multiplicative set consisting of the non-negative powers of $t$. But I do not know the maximal ideals of ...
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1answer
79 views

Is this ideal prime? [closed]

Let $A = k[X, Y, Z]/(XY - Z^2)$, where $k[X, Y, Z]$ is a polynomial ring over a field $k$. Let $x, y, z$ be the image of $X, Y, Z$ respectively by the canonical homomorphism $\phi\colon k[X, Y, Z] ...
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1answer
97 views

exercise of Matsumura about CM

I have 2 question about this exercise of Matsumura: question 1- why $y^3$ is $R/(x^3)$ regular? question 2- I hardly (in 20 lines) can prove is there a short way or intuition for this part ? ...
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1answer
43 views

exercise of Matsumura

my question is about this exercise of Matsumura: in the proof hint we use is this obvious? or e should define an isomorphism?
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1answer
63 views

History of five lemma

I am interested in the history of five lemma. Who was first to prove it and What was the purpose of proving it ? http://en.wikipedia.org/wiki/Five_lemma
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0answers
51 views

Is the length of the composition series of a free module identical to the number of its bases?

Let $A_0$ be an Artinian ring, $M$ a free $A_0$-module. Then, is the length of the composition series of $M$ identical to the number of its bases? It seems to me that it is not. If $\mathfrak a$ ...
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1answer
45 views

If $S$ is the integral closure of $R$ in it's field of fractions and $S\subset R_{m}$ is $R_{m}$ integrally closed?

Let $R$ be a domain and $K$ be it's field of fractions. Let $S$ be the integral closure of $R$ in $K$. Let $M$ be a maximal ideal of $R$. If $S\subset R_{M}$ is $R_{M}$ integrally closed in $K$? My ...
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0answers
57 views

Completion of integral domain

Let $A$ be an integral domain with the $I$-adic filtration. Let $B$ be the fraction field of $A$. My question is the following: Is the fraction field of the completion of $A$ the same as the ...
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1answer
60 views

Can a ring isomorphism change the structure of a module?

Let $M$ be an $R$-module, where $R$ is a ring with unit. Given a ring automorphism $\phi: R \rightarrow R$, we can define a new $R$-module structure on $M$ by $r \cdot x = \phi(r) x$ for all $r \in ...
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2answers
120 views

Coordinate ring of the unit circle is never a UFD?

I'm reading some notes about coordinate rings. On the third example on the second page, the author notes that the coordinate ring $K[\mathcal{C}]$ is not a UFD. If $f=X^2+Y^2-1$, then in ...
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1answer
46 views

Noetherian local ring, detail in theorem 1.3.16 in Liu

I can't understand a detail in the proof of theorem 1.3.16 in Liu. The theorem is: let $(A,\mathfrak{m})$ a Noetherian local ring, $\hat{A}$ its $\mathfrak{m}$-adic completion, $(B,\mathfrak{n})$ an ...
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1answer
27 views

Are the generators of the subgroup defining tensor products linearly independent over $\mathbb Z$?

Let $S$ be a (commutative) ring with identity, and let $M$, $N$ be $S$-modules. (I guess if $S$ isn't commutative, I want $M$ to be a right $S$-module an $N$ a left $S$-module.) In the definition of ...
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1answer
77 views

Number of maximal and prime ideals

Find how many prime and maximal ideals there are in the ring consisting of matrices $$M= \begin{bmatrix} a & b & c \\ 0 & a & b \\ 0 & 0 & a \\ \end{bmatrix} $$ ...
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2answers
84 views

Is quotient of a ring by a power of a maximal ideal local?

Say I have a commutative ring $R$ with a maximal ideal $m$. Then $m/m^k$ is a maximal ideal in $R/m^k$ for any $k$. Is it the only maximal ideal, i.e. is $R/m^k$ a local ring? This is a well ...
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2answers
96 views

$\mathbb{Q}[X,Y]/(X^2+Y^2-1)$ is integrally closed

I'm was browsing this question, where it is proven the quotient field of $\mathbb{Q}[X,Y]/(X^2+Y^2-1)$ is isomorphic to the rational function field $\mathbb{Q}(t)$ under the isomorphism $$ (x,y) ...
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0answers
57 views

$A'_{m'}$ is a finitely generated $A_{m}$-module?

Let $A$ be a finitely generated $k$-algebra that is a domain. Let $A'$ be the integral closure of $A$ in $\operatorname{Frac}(A)$. By finiteness of integral closure $A'$ is a finitely generated ...
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1answer
63 views

help in an example.

In page 226 of David Eisenbud's book Commutative Algebra with a View Toward Algebraic Geometry there is an example which I need help in some parts of it: why $codim I= 1$? why $dim M = dim R = ...
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1answer
37 views

What is the unmixedness theorem?

Let $R$ be a Cohen-Macaulay ring. Then for every ideal $I \subset R$, $p\in\operatorname{Ass}_{R}(R/I)$, $\operatorname{ht}(p)=\operatorname{ht}(I)$. This is the unmixedness theorem? I want to ...
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0answers
45 views

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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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 ...
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2answers
98 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 ...
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1answer
48 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
42 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 ...
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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 ...
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1answer
45 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
76 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
33 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 ...
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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 ...
3
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1answer
49 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 ...
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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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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 ...
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0answers
75 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
66 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 ...
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1answer
59 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 ...
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1answer
46 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
133 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 ...
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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 ...
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1answer
153 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 $<$ ...
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0answers
50 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
47 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
61 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
42 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 ...