1
vote
1answer
82 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
113 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
254 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 ...
-1
votes
1answer
45 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
74 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
63 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
76 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 ...
3
votes
0answers
70 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 ...
3
votes
1answer
38 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
30 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
112 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
36 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
18 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 $\dim R=0$?
6
votes
1answer
65 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 ...
1
vote
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
108 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
42 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 ...
0
votes
1answer
56 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 ...
5
votes
2answers
210 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 ...
-1
votes
1answer
40 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
17 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
28 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
42 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
37 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 ...
3
votes
1answer
74 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 ...
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!
1
vote
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?
3
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 ...
0
votes
0answers
38 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 ...
2
votes
1answer
166 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 ...
3
votes
1answer
48 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
35 views

Depth zero module and $R$-regular element

Let $(R,m)$ be a commutative Noetherian local ring with $\operatorname{depth}(R)>0$ and $M$ be a finitely generated $R$-module with $\operatorname{depth}(M)=0$. Then can we take an $R$-regular ...
1
vote
0answers
69 views

Cohen structure theorem [closed]

Let $R$ be a one-dimensional noetherian complete local ring and $I$ an ideal of it. Are there any conditions that satisfy $R\cong S/J$, where $S$ is a regular local ring with ideal $J$, such that $J$ ...
3
votes
1answer
82 views

If $\mathbb{Z}$ satisfies an identity $\eta$, then every **commutative** ring satisfies $\eta$? And related questions.

Assume all rings have unity and that ring homomorphisms preserve unity. Now by general principles, if every free object in the category of rings satisfies an identity $\eta$, then every object in the ...
3
votes
1answer
50 views

Use of Zorn's Lemma in showing nilradical equals intersection of primes

I'm very confused as to the use of Zorn's lemma in showing that the nilradical of a ring is the intersection of all the prime ideals. Namely, we let $a \notin N$, where $N$ is the nilradical. Then we ...
0
votes
1answer
43 views

Localization of a direct product

Is the localization of a direct product of two rings at a maximal (or prime) ideal identified with a localization of one of them? I would appreciate for any detailed answer.
2
votes
1answer
90 views

The family of schemes $\operatorname{Spec} A[x]/(x^n)$

Consider the family $S_n:=\operatorname{Spec} A[x]/(x^n)$ of schemes, $A$ denoting any ring (which in our subject always means commutative and with identity). Is there some intuitive picture for ...
1
vote
0answers
24 views

Why is $A/I^n A \cong \hat{A}/I^n \hat{A}$?

Let $A$ be a commutative ring, $I \subset A$ a finitely generated ideal. Define $\hat{A} := \varprojlim A/I^n$. What is the best way to proof that $A/I^nA \cong \hat{A}/I^n \hat{A}$ for all $n$?
0
votes
2answers
30 views

Question related to integrality of field of fractions

This is actually not a problem, but it's a statement which is taken for granted and I don't know how to prove it. Hope some one can help me. I really appreciate: Suppose $A$ is subring of ...
3
votes
2answers
45 views

ACC on principal ideals implies factorization into irreducibles. Does $R$ have to be a domain?

I am following an argument in chapter zero of Eisenbud's Commutative Algebra book. It is not clear whether or not he is assuming that $R$ is a domain. If I start the proof assuming $R$ is not ...
2
votes
3answers
115 views

$S^{-1}A \cong A[x]/(1-ax)$ [duplicate]

If $A$ is a commutative ring with unit, $a \in A $ and $S = \lbrace a^n \mid n \geq 0 \rbrace $ then there is an isomorphism $$S^{-1}A \cong A[x]/(1-ax).$$ In fact we can consider the ...
0
votes
1answer
58 views

If $R/I$ satisfies Serre's condition $S_2$ then $R/(I,x)$ satisfies $S_1$

Let $I$ be an ideal of polynomial ring $R=K[x_1,\ldots,x_n]$ and $x$ be a non-zero divisor of $R/I$. Is the following statement true? If $R/I$ satisfies Serre's condition $S_2$ then $R/(I,x)$ ...
1
vote
1answer
91 views

Morphism of rings and localization

Let $ \varphi : A \to B $ be a morphism of rings. Why are the two following assertions equivalent: $ 1) $ There exists a multiplicative subset $ S $ of the ring $ A $, and an ideal $ I $ of $ A $, ...
3
votes
1answer
47 views

A local PID is a Euclidean domain

Studying commutative algebra I've encountered this statement: A PID which is also a local ring is a Euclidean domain. Is it true ? Why ?
1
vote
1answer
47 views

Commutative ring can be homomorphically mapped onto field

During my algebra lecture, my lecturer used the fact that any commutative ring can be homomorphically mapped onto a field. Is the statement true? How to show that? Thanks