4
votes
1answer
53 views

Are $R=K[a,b,c,d]/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K[x^4,x^3y,xy^3,y^4]$ isomorphic?

I saw somewhere that $R=K[a,b,c,d]/(ad-bc, a^2c-b^3, bd^2-c^3, ac^2-b^2d)$ and $K[x^4,x^3y,xy^3,y^4]$ considered the same. Is it true? Why? I'm a beginner so please answer in details
1
vote
2answers
49 views

Show module is Noetherian

Let $R$ be a commutative ring and let $0 \to L \to M \to N \to0$ be an exact sequence of $R$-modules. Prove that if $L$ and $N$ are noetherian, then $M$ is noetherian. I tried considering the pre ...
1
vote
2answers
47 views

Finitely many prime ideals lying over $\mathfrak{p}$

Let $A$ be a commutative ring with identity and $B$ a finitely generated $A$-algebra that is integral over $A$. If $\mathfrak{p}$ is a prime ideal of $A$, there are finitely many prime ideals $P$ ...
0
votes
1answer
24 views

Integral multiplicative system over a domain

Suppose $A$ is a domain and $S\subseteq A$ is a multiplicative system. Show that $S\subseteq A^\times$ if and only if $S^{-1}A$ is integral over $A$. I've started $\Leftarrow$ below... Suppose ...
1
vote
1answer
49 views

Necessary and sufficient condition for $r(\mathfrak a)$ to be prime

As we know, $$\mathfrak a~\text{is a primary ideal}\Rightarrow r(\mathfrak a)~\text{is a prime ideal}. $$ But $r(\mathfrak a)$ may not be a prime ideal if $\mathfrak a$ isn't a primary ideal. ...
0
votes
2answers
61 views

Examples of Cohen-Macaulay rings.

I've just started to learn about Cohen-Macaulay rings. I want to show that the following rings are Cohen-Macaulay: $k[X,Y,Z]/(XY-Z)$ and $k[X,Y,Z,W]/(XY-ZW)$. Also I am looking for a ring which is ...
5
votes
0answers
45 views

Is there a characterization of integral domains in terms of the homomorphisms out of them?

In the $\mathbf{Set}$-concrete category of commutative rings, we can define that an object $A$ is a field iff for every homomorphism $f : A \rightarrow B$, precisely one of the following holds. $f$ ...
1
vote
1answer
68 views

Every radical is prime?

$a$ is an ideal of $A$. $$f:A\to A/a,\ \ x∈r(a)$$ r(a) is a prime ideal? proof 1: $x^n\in a$ for some $n \Rightarrow (x+a)^n\in a$ for some $n \Rightarrow f(r(a))=\text{nil-radical}$ in $f(a) ...
1
vote
0answers
51 views

Prove this morphism is a derivation

Let $A$ a ring, $B$ a $A$-algebra, $I=\ker (B\otimes_A B \to B)$ given by multiplication, i wanna see why is $$d:B \to I/I^2$$ $$b \mapsto b\otimes 1 -1 \otimes b$$ well defined (in fact, why mod ...
1
vote
0answers
48 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 ...
1
vote
1answer
97 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 ?
0
votes
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 ...
2
votes
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 ...
0
votes
1answer
22 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 ...
3
votes
2answers
82 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 ...
0
votes
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 ...
0
votes
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.
0
votes
2answers
75 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 ?
2
votes
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
votes
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 ...
1
vote
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 ...
1
vote
0answers
73 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 $I=\mathfrak{p}_1\cap \mathfrak{p}_2$, ...
1
vote
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.
0
votes
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
votes
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 $a\otimes b$. ...
0
votes
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
votes
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?
2
votes
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 ...
1
vote
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 ...
3
votes
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
votes
0answers
49 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 $p \in \operatorname{Ass_{R}}(R/I)$, $\operatorname{ht}(p)= \operatorname{ht}(I)$?
1
vote
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 ...
0
votes
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 ?
1
vote
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) ...
0
votes
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?
2
votes
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 ...
2
votes
1answer
39 views

If ring $B$ is integral over $A$, then an element of $A$ which is a unit in $B$ is also a unit in $A$.

Let $A$ be a subring of ring $B$, with $B$ integral over $A$. If $x$ in $A$ is a unit in $B$, then it is a unit in $A$. I know that $f(t) = t - x$ is in $A[t]$ with $f(x) = 0$, and that there ...
5
votes
1answer
54 views

Topological closure of ideal in $A[[T]]$ - Proposition 1.3.7 in Liu

In Proposition 1.3.7 of Liu's book, one proves that if a ring $A$ is noetherian then so is $A[[T]]$. We take an ideal $I$ of $A[[T]]$ and prove that there exist $F_1,\ldots,F_m\in I$ such that for all ...
0
votes
1answer
40 views

Every irreducible element is prime: always holds under surjective ring homomorphism?

Let $f:R\rightarrow S$ be a surjective ring homomorphism. $R,S$ are both integral domains. If every irreducible element in $R$ is prime, then is it true that every irreducible element in $S$ is ...
2
votes
2answers
41 views

Ring homomorphism with field as image, is the pre-image also a field?

Let $f:R\rightarrow S$ be a surjective ring homomorphism. $R,S$ are both integral domains. Suppose $S$ is a field, then is $R$ also a field? A possible useful fact: A finite integral domain is a ...
1
vote
0answers
25 views

Grading and commutators

If $R$ is a unital associtaitve commutative ring, then can any $R$-algebra $A$ may be filtered as $A_0:=A$ and $A_{i+1}:=(A_i,A_i)$ where $(-,-)$ is the commutator of $A$ with respect to $A$'s ...
2
votes
0answers
91 views

Associated primes and their heights

Let $(S,m)$ be a commutative Gorenstein local ring, $I$ an ideal of $S$ such that $\operatorname{ht} I=t$, and $R=S/I$. Let $a \in m$ be an $R$-regular element such that for any prime ideal ...
1
vote
1answer
43 views

Prime ideals in formal power series

Let $A$ be a commutative ring with unit. If $\mathfrak{p} \subset A $ is a prime ideal, then $\mathfrak{p}$ is the contraction of a prime ideal of $A[[x]]$, the ring of formal power series. Why is ...
7
votes
1answer
114 views

What's the motivation of definition of primary?

Primary ideal can be regard as the generalization of prime ideal and radical. But Why it's defined like that?It's not symmetry. Why not define like that:
2
votes
1answer
86 views

Question about some details of a proof

i) Why it's a unit can prove this proposition ii)see picture
-2
votes
1answer
66 views

Surjective Implies Injective for R-Homomorphism on Finitely Generated Module [duplicate]

Let $M$ be a finitely generated module over a ring $R$, and let $f$ be an $R$-homomorphism from $M$ to itself. Does $f$ injective imply $f$ surjective? Does $f$ surjective imply $f$ injective? I have ...
0
votes
1answer
30 views

Finitely generated ideal in boolean ring [duplicate]

A boolean ring is a commutative ring where $x^{2} = x$ for every $x$. Why in such a ring a finitely generated ideal is principal ?
4
votes
3answers
200 views

Idempotent in a local ring

Is it true that a local ring, i.e. a commutative ring with a unique maximal ideal, doesn't contain idempotent elements $\neq 0, 1$ ? Why ? Any hint ?
1
vote
2answers
64 views

Zero dimensional local ring with maximal ideal not principal.

Probably it is well known. I am looking for a zero dimensional local ring with maximal ideal not principal.
1
vote
0answers
45 views

Are Ideals and Varieties Inclusion Reversing?

Let $S_1$, $S_2$ be sets or varieties (I don't think it matters, does it?). Then if $S_1 \subset S_2$, is it always the case that $I(S_2) \subset I(S_1)$ (where I is an ideal)? Also, is it always the ...