0
votes
1answer
26 views

Example of commutative algebra over integers where there exists $x$ such that $x = y^2$ for several $y$'s

Is there a commutative algebra over integers such that there exists $x$ with $x = y^2$ for several $y$'s? Also, is there a commutative algebra over integers such that for every $k \in \mathbb{N}$, ...
1
vote
1answer
29 views

Ideal quotient and extension

Let $R$ be a commutative ring and $S$ a subring of $R$. If $I$ is an ideal of $S$ define $I^e$ as the ideal in $R$ generated by $I$, i.e. the extension of $I$ in $R$. If $I,J$ are ideals in $S$, we ...
0
votes
1answer
52 views

Proof that presheaf is a sheaf for Spec

Atiyah Macdonald define presheaf (chapter 3, exercise 23) on the base of $Spec(A)$, where $A$ is commutative ring with $1$, as follows $$ \mathfrak{F}(X_f) = A_f, $$ where $X_f$ is a basic open set ...
1
vote
2answers
137 views

Every element in a ring with finitely many ideals is either a unit or a zero divisor.

I came across the above proposition on mathstackexchange If every nonzero element of $R$ is either a unit or a zero divisor then $R$ contains only finitely many ideals. the link asks a different ...
1
vote
1answer
78 views

The spectrum of a commutative ring with unity and its “topology”

Let $\operatorname{Spec}(R)$ be the set of prime ideals in the commutative ring with unity $R$, and let $\mathfrak a$ be some ideal. Show that we get a topological space if we define the closed sets ...
2
votes
1answer
44 views

Von Neumann regular but not self-injective ring

I want an example of a von Neumann regular ring which is not self-injective. My thanks go to anybody answering.
7
votes
2answers
178 views

Showing that $x^3+y^3+z^3=0$ is not rational

Is there a short proof that $F:x^3+y^3+z^3=0$ in $\mathbf{P}^2$ is not rational, apart from using the genus? Perhaps this is an elliptic curve, so every morphism $\mathbf{P}^n\rightarrow F$ is ...
2
votes
1answer
32 views

Example of a ring which is not CM at all its prime ideals

A commutative ring $A$ is said to be CM at a maximal ideal $\mathfrak{m}$ if and only if $Depth(A_{\mathfrak{m}})=Krull(A_{\mathfrak{m}})$. What is an example of a connected commutative ring $A$ which ...
0
votes
1answer
46 views

A nonregular local ring [duplicate]

Consider the ring of the formal power series $k[[T_1,\ldots,T_n]]$ ($k$ algebraically closed) where $\mathfrak m$ is the maximal ideal. If $f\in\mathfrak m^2$, why $$\frac{k[[T_1,\ldots,T_n]]}{(f)}$$ ...
0
votes
1answer
71 views

Some residue field

Consider a prime ideal $\mathfrak{p}\in\mathrm{Spec} \ \mathbf{Z}[x]$; the residue field at $\mathfrak{p}$ is the fraction field of $\mathbf{Z}[x]/\mathfrak{p}$. Can we classify the residue fields? I ...
0
votes
0answers
101 views

Multiplicity of a module in a particular case

We define multiplicity of a module M of dimension $d>0$ as $$mult(M) := lc (P_M) (d-1)!$$ where $P_M$ denotes the Hilbert polynomial of M. Equivalently, we have $mult(M) = Q_M(1)$, where $HP_M (z) ...
0
votes
2answers
59 views

Cohen-Macaulay and regularity

I know this is a simple question but to make sure....: $A$ is a commutative ring which is Cohen-Macaulay and for every maximal ideal $\mathfrak{m}$ in $A$ if $\dim A_{\mathfrak{m}}=\dim A$ then ...
3
votes
2answers
38 views

Kahler differentials and quotient rings.

I am dealing with some nice rings that are always isomorphic to some fairly nice quotient ring of a polynomial ring. A typical example is: $$ \mathbb{C}[X,XY,XY^2] \cong ...
1
vote
0answers
50 views

Krull dimension of localization

If $R$ is a commutative ring and $m$ a maximal ideal therein, then what are the conditions for the Krull dimension of $R$ equaling to the Krull dimension of $R_m$?
1
vote
3answers
100 views
+50

Some practical questions on cohomology and the ring $\mathbf{Z}[x]/(x^2)$

So I know that the cohomology ring of $S^n$ is $\mathbf{Z}[x]/(x^2)$ with "$x$ in degree $n$"; if we ignore the grading then this ring fails to distinguish the spheres. What is actually meant by "in ...
0
votes
1answer
68 views

Canonical ring map

Let $\chi:\mathbf{Z}\rightarrow A$ be the canonical map to a ring $A$, and let $p$ be a prime ideal of $A$. Then I claim that $\chi^{-1}(p)=(\mathrm{char} \ k(p))$ where $k(p)$ is the residue field at ...
1
vote
2answers
41 views

Finitely generated ideal with special property

Is there a ring with a finitely generated ideal $I$ which has an infinite subset $M\subseteq I$ such that $M$ generates $I$ but no finite subset of $M$ does it? What I found out: If such a rings ...
3
votes
1answer
49 views

How to characterize all finite commutative local rings with the maximal ideal of fixed order (if the order is small)?

Let $R$ be a finite commutative local ring with the maximal ideal $M$ of order $m$. How to characterize all such finite commutative local rings? For examples, if $m=2$, then $R\cong\mathbb{Z}_4$ ...
1
vote
1answer
51 views

Injectivity of simple modules

If $R$ is a commutative ring with $1$ having a maximal ideal $m$ such that the local ring $R_m$ is a field, how could one check that $R/m$ is an injective $R$-module? If we want to use Baer Lemma, we ...
2
votes
2answers
76 views

When a finite local ring $R$ has $-1$ as a square in $R^\times$?

Let $R$ be a finite local ring with maximal ideal $M$ such that $|R|/|M|\equiv 1\pmod{4}$. Then $-1$ is a square in $R^\times$ (that is, there exists $u\in R^\times$ such that $u^2=-1$) if and only ...
0
votes
1answer
62 views

Rank of a module when the base ring is not a domain

Suppose $R$ is a commutative Noetherian local ring with $1$, which is not a domain. Let $M$ be a (non-free) finite $R$-module. What is meant by rank of $M$ in this case?
0
votes
1answer
35 views

Does the relation $\pi(S_{i})=S^{-1}R-P_{i}\cdot S^{-1}R$ hold for prime ideals $P_i$ in a commutative ring $R$?

Let $R$ be a commutative ring. Let $P_{i}$, $1\leq i\leq n$ be prime ideals none of which are contained in each other. Let $S=R-(\cup_{i=1}^{n} P_{i})$. Then $S$ is a multiplicatively closed set and ...
1
vote
1answer
86 views

How do we know that $f(x)\in Y$?

At page 19 in this book $f:X\to Y$ is defined to be $$f(a):=(\tilde\varphi(T_1')(a),\dots,\tilde\varphi(T_n')(a)).$$ To explain the notation above, $X\subseteq \mathbb{A}^m(k)$, $Y\subseteq ...
1
vote
2answers
135 views

Is $\mathbb{C}[x,y] / (y^2-x^3)$ a PID?

First, I'd like to show $\mathbb{C}[x,y] / (y^2-x^3)$ is an integral domain. Then I need to find out whether or not it is a PID. For the first part, I want to show $y^2-x^3 \: | \: fg \implies ...
6
votes
5answers
302 views

A finite commutative ring with 1 whose elements satisfy a particular equation

I would be very grateful if you give me a hint on it: Suppose $R$ is a finite commutative ring with identity such that $ x^3 = x $ for all elements $x$ of $R$. Then $R$ is a finite direct product ...
3
votes
2answers
75 views

How to show this ring is not a UFD

I am trying to show that $R=\Bbb Z[x,y,z,w]/(xw-zy)$ is not a UFD. Let $I=(xw-zy)$. Let $X=x+I$, $Y=y+I$, $Z=z+I$, and $W=w+I$. My guess is that $X$ is irreducible and therefore $(X)$ is a ...
-1
votes
1answer
48 views

Graded ring, and its homogeneous ideals : $ I = \displaystyle \bigoplus_ {n \in \mathbb {Z}} (I \bigcap B_n) $

Let $ B = \displaystyle \bigoplus_{n \in \mathbb {Z}} B_n $ be a graded ring. Let $ I $ be an ideal of $ B $. Why is $ I = \displaystyle \bigoplus_ {n \in \mathbb {Z}} (I \bigcap B_n) $ equivalent to ...
5
votes
3answers
83 views

Finitely generated ideal in Boolean ring; how do we motivate the generator?

This problem is Exercise 11.3 in Atiyah/Macdonald Commutative Algebra. They ask to prove every finitely generated ideal in a Boolean ring is in fact a principal ideal. The question has been answered ...
0
votes
1answer
36 views

Global dimension regular rings of finite type

Have I made an error in my reasoning? If $k$ is a field, $A$ is a commutative regular $k$-algebra of finite type and ${\mathfrak{m}}$ is a maximal ideal in $A$ then since $Ext_{A_{\mathfrak{m}} ...
0
votes
0answers
43 views

Is every local ring the localization of some other ring?

One way of constructing a local ring is to start with any commutative ring, and localize all the elements outside of some maximal ideal (i.e., adjoining inverses to all those elements). But I'm ...
5
votes
1answer
64 views

Irreducibility of some multivariate polynomials

Consider the polynomials $xw-yz\in A[x,y,z,w]$ and $x^n+y^n+z^n\in A[x,y,z]$, where $A$ is a commutative ring. I am curious to know what conditions on $A$ (factorial ring, algebraically closed field, ...
0
votes
2answers
42 views

$R^{(I)} \cong K \oplus H$ where $R^{(I)}$ is free but $K$ is not free

Let $R$ be a commutative ring with unit. Is there an example of a direct sum of $R$-modules $$R^{(I)} \cong K \oplus H$$ where $R^{(I)}$ is free but $K$ is not free ? Clearly $R$ can't be a PID.
1
vote
0answers
82 views

What is $\operatorname{Ass}\operatorname{Ext}^i(M,N)$?

This is exercise 1.2.27 of Bruns-Herzog: Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $N$ an arbitrary $R$-module. Deduce that $\operatorname{Ass}(\operatorname{Hom}_R(M,N)) = ...
1
vote
1answer
45 views

Localizations of $ \mathbb{Z}_{p^k}$

Let $S \subseteq \mathbb{Z}_{p^k} $ be a multiplicative subset, where $p$ is a prime number, $k$ an integer. Is it true that $$S^{-1} \mathbb{Z}_{p^k} \cong \mathbb{Z} /n\mathbb{Z} $$ for some ...
2
votes
1answer
33 views

Congruence in localization of rings

Please help me to prove for all maximal ideals $\mathfrak{m}$ of $R$, $(aR/a^2R)_\mathfrak{m}\cong (aR)_\mathfrak{m}/(a^2R)_\mathfrak{m}\cong aR_\mathfrak{m}/a^2R_\mathfrak{m}$, where $R$ is a ...
2
votes
0answers
71 views

Automorphism of certain f.g. free modules

This is a quick question from Frohlich and Taylor's Algebraic Number Theory, II.4, p 94. Let $R$ be a Dedekind domain with quotient field $K$, $\mathfrak p$ is a non-zero prime ideal of $R$ and ...
2
votes
1answer
43 views

Prime radical that is nil but not nilpotent

Please help me to show that the prime radical of the ring $R=\prod\limits_{n = 1}^\infty { \mathbb{Z} /2^n\mathbb{Z} } $ is nil but not nilpotent.
1
vote
1answer
32 views

Basis for the completion of a free module

This (or similar) question might have been asked before- apologies for any duplication. I've got a Dedekind domain $R$, a non-zero prime ideal $P$ of $R$ and the completion $\widehat{R}$ of $R$ wrt ...
1
vote
1answer
117 views

Localization and direct limit

Let $ A $ be a ring. Let $ I $ be a preordered set, filtering. Let $ \Sigma $ a multiplicative subset of $ A $. Suppose for any given $ i \in I $ a multiplicative subset $ S_i $ of $ A $ contained ...
2
votes
1answer
44 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) ...
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 $\dim R=0$?
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 ...
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
0answers
44 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
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 ...
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 ...
-3
votes
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
votes
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 $K$ ...