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

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2
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1answer
42 views

In arbitrary commutative rings, what is the accepted definition of “associates”?

In an integral domain, the following are equivalent: $r \mid s$ and $s \mid r$ $r=us$ for some unit $u$ However in arbitrary commutative rings this is no longer the case; in particular, (2) ...
2
votes
1answer
74 views

Does $\operatorname{Hom}(M,T)\cong\operatorname{Hom}(N, T)$ for all $A$-modules $T$ mean $M\cong N$?

The question is contained in title, I'm working with $A$-modules $M$ and $N$. I feel like Yoneda's lemma is what I'm looking for but it applies to functors into the category of sets, whereas ...
0
votes
0answers
78 views

Atiyah & Macdonald's Introduction to Commutative Algebra, Exercise 8.5

The exercise asks the reader to prove that $X$ is a finite covering (i.e., the number of points of $X$ lying over a given point of $L$ is finite and bounded) of $L$, where the affine varieties $X$ and ...
3
votes
1answer
74 views

How to prove this innocent looking isomorphism

I've got a Dedekind domain $R$ with quotient field $K$, a non-zero prime ideal $P$ of $R$. I form the completion $\widehat{K}$ of $K$ wrt the valuation $v_P$ associated to $P$. Let $\widehat{R}$ be ...
0
votes
0answers
19 views

Relation between von Neumann regular rings, Krull dimension 0, and rings with no nonzero nilpotents. [duplicate]

Why a ring $R$ is von Neumann regular if $R$ has no nonzero nilpotents and $\dim R=0$?
1
vote
1answer
45 views

Support of a quasicoherent sheaf

When $M$ is a finitely generated module over a commutative ring $R$, it is easy to see that the support of $\tilde{M}$ on $\mathrm{Spec}\,R$ is given by $V(\mathrm{ann}_R(M))$. This is not true for ...
4
votes
1answer
70 views

Help with a problem from Christian Peskine's book about Artinian rings

I am stuck with this problem from the book of Complex Projective Geometry. Let $A$ be a Noetherian ring. Assume that if $a \in A$ is neither invertible nor nilpotent, then there exist $b \in A$ such ...
5
votes
2answers
96 views

Localization does not commute canonically with infinite direct products

Let $S=\mathbb{Z}-\{0\}$, and the fraction ring \begin{equation} S^{-1}\prod_{1}^{\infty}\mathbb{Z}_{i}=\{\frac{(a_{1},a_{2},...,a_{n},...)}{b}:b,a_{i}\in\mathbb{Z},b\neq 0\}.\end{equation} Show ...
0
votes
0answers
35 views

Integral dependence of coordinate ring

In Hartshorne P18-P19, the proof of Thm. 3.4 shows that the ring $S(Y)_{(x_{i})}$ is contained in the integral closure of the coordinate ring $S(Y)$ (all regarded as subrings of the quotient field of ...
2
votes
0answers
68 views

Direct product of direct sum of a flat module

I have a problem concerning flat modules: Let $M$ be an $R$-module such that the direct product $M^A$ is flat for all sets $A$. I want to prove that $(M^{(B)})^A$ is also flat for any sets $A$ ...
0
votes
1answer
60 views

Irreducible components in the spectrum of a ring

I have a question concerning page 43 of this book. In Corollary 2.7 it says that the map $\mathfrak{p}\mapsto \overline{\{\mathfrak{p}\}}$ is a bijection from Spec($A$) onto the sets of closed ...
-2
votes
1answer
59 views

Krull dimension of $\mathbb{C}[x,y,z]/I$ where $I=(x^2-yz,xz-x)$.

Krull dimension of $\mathbb{C}[x,y,z]/I$ where $I=(x^2-yz,xz-x)$. The problem says first verify $p_1=(x,y)$, $p_2=(x,z)$ and $p_3=(x^2-y,z-1)$ are prime minimal over $I$. How can I use it ?
1
vote
2answers
71 views

Help with $\sqrt{I}$, where $I=(y^2,x+yz)$ in $\mathbb{C}[x,y,z]$

$a)$ $\sqrt{I}$ where $I=(y^2,x+yz)$ in $\mathbb{C}[x,y,z]$. first it's clear $y \in \sqrt{I}$ then $x=(x+yz)-yz \in \sqrt{I}$ because $yz \in \sqrt{I}$ is it $\sqrt{I}=(x,y)$ ? $b)$ ...
0
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1answer
37 views

Find the height of prime ideal $p=(x_n-x_1^n,\ldots ,x_2-x_1^n)$ in $\mathbb{C}[x_1,\ldots,x_n]$

Find $\operatorname{ht}(p)$ where $p=(x_n-x_1^n,\dots,x_2-x_1^n)$ ideal of $\mathbb{C}[x_1,\ldots,x_n]$. $\operatorname{ht}(p)=$ height of a prime $p$ How to prove $p$ is prime ?
2
votes
1answer
57 views

In $\Bbb Z[x,y]$ is $(x^2+1,y^2+1,-xy+1)$ prime?

This is a reality check for the following computations that I did: Consider the map $(\operatorname{id}, \iota): \Bbb A_\Bbb Z^1 \rightarrow \Bbb A_\Bbb Z^1\times \Bbb A_\Bbb Z^1$ from the definition ...
-1
votes
1answer
30 views

Noether normalisation $A=\mathbb{C}[x,y]/(f)$ where $f=(x-a)y^2-(x-b)$ find a transcendence element

Noether normalisation $A=\mathbb{C}[x,y]/(f)$ where $f=(x-a)y^2-(x-a)$ $a , b \in \mathbb{C}$ find $z \in A$. transcendence over $\mathbb{C}$ such that $A$ is integral over $\mathbb{C}[z]$ any ...
4
votes
2answers
125 views

Proving that tensoring a projective module with a flat module gives a projective module?

If $P$ is a projective module and $M$ is a flat module, both over some commutative ring $R$, then how do you prove that $M\otimes_R P$ is flat? I tried using the fact that $P$ is the direct ...
0
votes
1answer
71 views

Prove the ideal $(f)$ is not maximal

I'm trying to solve the following problem: Let $B$ be a UFD and $A := B[y]$ the polynomial ring. Let $f$ be a polynomial that has a term $by^i$ with $i > 0$ such that $b$ is not divisible ...
0
votes
0answers
24 views

$ \mathrm{Spec} ( A \times B ) = \mathrm{Spec} A \coprod \mathrm{Spec} B $ [duplicate]

Let $ A $ and $ B $ be two commutative rings. Why is : $ \mathrm{Spec} ( A \times B ) = \mathrm{Spec} A \coprod \mathrm{Spec} B $ ?. Thanks a lot.
3
votes
0answers
109 views

Counterexamples for lcm-gcd identity and modular law for rings

In Miles Reid's Undergraduate Commutative Algebra, Exercise 1.3, we need to find counterexamples of lcm-gcd identity and modular law in the ring $A=k[X,Y]/(XY)$: $(I+J)(I\cap J)=IJ$; ...
2
votes
1answer
47 views

Transcendence degree of fraction field

Let $k$ be a field and $p \in k[x_1, \dots, x_n]$ an irreducible element. Is there an elementary way to prove that $\operatorname{tr.deg}_k \mbox{Frac}(k[x_1, \dots, x_n]/(p)) = n-1$?
2
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0answers
43 views

A question about the proof of Hilbert's Basis Theorem

I have a question regarding the proof of Hilbert's Basis Theorem. Say $I=(f_1,f_2,f_3,\dots)$ is an ideal in $A[x]$, where A is a Noetherian ring. Say we take the leading coefficients $a_i$ of all ...
1
vote
1answer
31 views

Integral dependence and field extension

Let $R$ be a domain (commutative with unity). $k$ is field algebraically dependent on $k_0$. $A$ is some ideal of $R \otimes_{k_0} k$ and $A_0$ = $A \cap R$. How to prove that $(R \otimes_{k_0} k)/A$ ...
-1
votes
1answer
57 views

A question related to associated prime ideals

Let $f:A\to B$ be a (commutative) ring homomorphism, $f^*:\operatorname{Spec}A\leftarrow\operatorname{Spec}B$ the induced map, and $N$ a $B$-module. It is well known that ...
0
votes
1answer
45 views

Characterization of Discrete Valuation Rings

Let $R$ be a Noetherian local domain with unique maximal ideal $M$. Then I want to show that if every $M$-primary ideal is a power of $M$, then $R$ is a Discrete Valuation Ring. I know I'll be ...
4
votes
1answer
39 views

Quotient $M/M^2$ is finite dimensional over $R/M$ in local Noetherian ring?

I have that $R$ is a Noetherian local ring with maximal ideal $M$, and I want to show that $M/M^2$ is a finite dimensional vector space over the field $R/M$. I think I've proved this (though I ...
1
vote
1answer
285 views

Isomorphism from $B[y]/IB[y]$ onto $(B/I)[y]$

For some reason I can't crack the following problem: Let $B$ be a ring, $I$ an ideal, and $A := B[y]$ the polynomial ring. Construct an isomorphism from $A/IA$ onto $(B/I)[y]$. How to ...
2
votes
2answers
47 views

Relation between $\operatorname{Coker}(f)$ and $\operatorname{Coker}(f \otimes 1_P)$

Let $M,N,P$ be $R$-modules ($R$ commutative ring with $1$) and let $f:M\to N$ be a $R$-module homormorphism. Let tensor the homomorphism to get $ f \otimes 1_P : M \otimes P \to N \otimes P $. I ...
3
votes
2answers
83 views

What kind of algebraic structure is this

I know that a commutative ring with an additional scalar multiplication on it is called an associative algebra. If the ring also has a 1 it is called a unital algebra. What would you call a field with ...
2
votes
1answer
62 views

Prove that if the induced homomorphism $M/\mathfrak aM \to N/\mathfrak aN$ is surjective, then $f$ it's surjective.

This problem it's from Atiyah and Macdonald, Chapter 2. Let $A$ be a commutative ring with $1 \ne 0$ and let $\mathfrak a$ be an ideal of $A$ contained in the Jacobson radical. Let $M$ be an ...
6
votes
2answers
213 views

What is an example of two k-algebras that are isomorphic as rings, but not as k-algebras?

Let $k$ be a field. Let $A$ and $B$ be two $k$-algebras, ie. two rings that are also $k$-vector spaces and their multiplication is $k$-bilinear. Any isomorphism of $k$-algebras is also a ring ...
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1answer
55 views

One dimensional noetherian domain

Let $(R,m)$ be a one-dimensional Noetherian domain. Is $R$ a regular or a topical ring like Gorenstein or other kinds?
0
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1answer
18 views

Finding a particular principal open subset of $Spec R$

Let $V\subseteq U$ be open subsets of $X=\text{Spec } R$, where $R$ is a commutative ring. So $V$ is the set of prime ideals not containing some ideal $I$, and $U$ is the set of prime ideals not ...
0
votes
2answers
29 views

Residue field of a local ring as field extension

Let $k$ be a field, $A$ a finitely generated, commutative $k$-Algebra and $\mathfrak p$ a prime ideal of $A$. Let $K$ be the residue field of the local ring $A_\mathfrak{p}$. I want to show that $K$ ...
1
vote
1answer
44 views

Question about completion of DVR.

Let $(R, (\pi))$ be a discrete valuation ring with residue class field $R/(\pi) \cong k$. It is well known that if $k$ embedds into $R$, then there is an isomorphism of the completion $\hat{R} \cong k ...
0
votes
1answer
38 views

A prime ideal in the intersection of powers of another ideal

Let $K$ be a field. Is it true that for any prime ideal $P$ of the ring $K[[x,y]]$ which lies properly in the ideal generated by $x$, $y$ we have $P⊆⋂_{n≥0}(x,y)^n$? My try is to choose the ...
2
votes
1answer
77 views

Exercise from Kaplansky's Commutative Rings and Eakin-Nagata Theorem

Exercise 15 of section 2-1 of Kaplansky's Commutative Rings is to show that if $T$ is a Noetherian ring and is finitely generated module over a subring $R$ of $T$, then $R$ is Noetherian. Kaplansky ...
2
votes
1answer
56 views

Possible examples where the Zero Divisor Conjecture does not hold

Given a ring $R$ with a nonzero zero divisor $x$, it is easy to show that if $M$ is a nonzero $R$-module, then there exists $y\in R-\{0\}$ such that $ym=0$ for some $m\in M-\{0\}$. I was ...
0
votes
1answer
44 views

tensor product of R-algebra and f.g module [closed]

$R$ is a commutative noetherian ring. If $S$ is an $R$-algebra, and $M$ a finitely generated $R$-module, is $M\otimes_RS$ finitely generated $S$-module? I only need a hint. Thanks!
0
votes
1answer
47 views

If the localization of a ring is a field, then the ring is an integral domain?

Let $R$ be a ring, and let $D$ be a multiplicatively closed subset of $R$. Is it the case that if $D^{-1}R$ is a field, then $R$ must be an integral domain?
2
votes
1answer
43 views

For some finitely many nonzero prime ideals, the contraction and extension of their product is zero

I was reading P.M. Eakin's thesis paper, The converse to a well known theorem on Noetherian Rings. The following is taken from Theorem 2, page 281 of that paper, and that's where I'm stuck. Let ...
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0answers
39 views

Prime ideal is contraction of prime ideal iff it's saturated

Let $\varphi: A\to B$ be a commutative ring homomorphism and $P$ a prime ideal of $A$. The expansion of an ideal $I\subset A$ is the ideal generated by $\varphi(I)$ in $B$, and the contraction of an ...
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1answer
37 views

2-dimensional Cohen-Macaulay domain

Any $1$-dimensional normal Cohen-Macaulay domain is a Dedekind domain. I am searching for a $2$-dimensional Cohen-Macaulay (normal or not) domain. Thanks in advance for any suggestion.
2
votes
1answer
169 views

Is this module noetherian?

Let $k$ be a field, and let $A$ be a commutative $k$-algebra. Assume that $A$ is a noetherian ring, and let $I\subseteq A$ be a proper ideal. Consider the ideal $I\otimes_k A \subseteq A\otimes_k ...
0
votes
1answer
37 views

A quotient of a regular local ring may not be regular

Let $(R,m)$ be a regular local ring having an ideal $I$ such that $I$ is a subset of $m^2$. If $I$ possesses a non-zerodivisor, I want to show that $R/I$ can not be regular. My try is just that ...
2
votes
1answer
51 views

Maximal linearly independent sets in a f.g. module

Suppose $M$ is a finitely generated module over a commutative unital ring $R$. Is it true that every maximal linearly independent set in $M$ has the same size? What is the most general condition ...
0
votes
1answer
53 views

Characterization of the kernel and cokernel of the natural homomorphism between a module and its double dual. [closed]

Let $R$ be a Noetherian ring and $M$ a finite $R$-module. Suppose $$ G \overset{\varphi}{\rightarrow} F \to M \to 0$$ is exact where $F,G$ are finite free modules. Suppose ...
0
votes
1answer
58 views

A finite module over a Noetherian ring is torsionless if and only if it is a submodule of a finite free module

Let $R$ be a Noetherian ring, and $M$ a finite $R$-module. Then $M$ is torsionless if and only if it is a submodule of a finite free module, where torsionless is defined here. (Bruns and Herzog, ...
2
votes
2answers
187 views

What to study from Eisenbud's Commutative Algebra to prepare for Hartshorne's Algebraic Geometry?

I surveyed commutative algebra texts and found Eisenbud's "Commutative Algebra: With a View Toward Algebraic Geometry" to be the most accessible for me. The book outlines a first course in commutative ...
5
votes
2answers
82 views

If $\{M_i\}_{i \in I}$ is a family of $R$-modules free, then the product $\prod_{i \in I}M_i$ is free?

If $\{M_i\}_{i \in I}$ is a family of free $R$-modules, then $\bigoplus_{i \in I}M_i$ is free. Is this true for the product $\prod_{i \in I}M_i$ too?