0
votes
1answer
32 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}} ...
5
votes
4answers
153 views

Euclid's proof of the infinitude of primes to prove this question

I'm trying to prove that if $k$ is a field, then there are an infinite number of irreducible monic polynomials in $k[X]$. My attempt of solution is use almost the same strategy of the Euclid's proof ...
5
votes
1answer
62 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, ...
1
vote
1answer
27 views

Submodules and quotients of free modules over Noetherian local rings

Let $R$ be a Noetherian local commutative ring, $F$ a finitely generated free $R$-module and $A,B$ some arbitrary $R$-modules. Consider a short exact sequence $0 \to A \to F \to B \to 0$. In [Bruns, ...
1
vote
1answer
66 views

Difference between algebraic and integral extension

I have been reading Miles Reid Undergraduate Commutative Algebra and in chapter 4 he talks about a crucial difference between algebraic extension and integral extension (see the picture below). Now I ...
0
votes
2answers
39 views

Every Artinian ring is isomorphic to a finite direct product of Artinian local rings

I was reading a proof of the above theorem (1.6.7 Theorem) from here, but there was something that confused me. The proof says $R$ has finitely many maximal ideals $M_1, \ldots ,M_r$, and the ...
0
votes
0answers
39 views

Exercise 7.10 Atiyah, $M[x] $ is a noetherian $A[x] $-module [duplicate]

The exercise is: Let $M$ be a noetherian $A$-module. Then $M[x] $ is a noetherian $A[x] $ module. The action of $A[x] $ on $M[x] $ is the obvious one. In a previous exercise it was shown that ...
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.
2
votes
1answer
56 views

$k[X,Y]/(f)$ not finitely generated as a module (Exercise 4.10 Reid, UCA)

I have been wrestling with this problem for some time and I still can't find $f$. It seems really simple, which annoys me even more. The problem is as follows (Exercise 4.10 Reid, UCA): Suppose ...
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 ...
3
votes
0answers
21 views

Localization and Direct limit [duplicate]

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 ...
3
votes
0answers
69 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 ...
13
votes
2answers
246 views

Quotient of polynomials, PID but not Euclidean domain?

While trying to look up examples of PIDs that are not Euclidean domains, I found a statement (without reference) on the Euclidean domain page of Wikipedia that $$\mathbb{R}[X,Y]/(X^2+Y^2+1)$$ is ...
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
106 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 ...
1
vote
1answer
41 views

If $A\ne 0$ is a square matrix over a commutative ring with $\det A=0$, then its null space contains an element whose components are minors of $A$

Let $R$ denote a commutative ring and $A\ne 0$ a $n\times n$ matrix over $R$ with $\det A=0$. Then there exists a $x\in\ker A\setminus\left\{0\right\}$ such that all components of $x$ are minors of ...
3
votes
1answer
69 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 ...
5
votes
2answers
93 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 ...
4
votes
2answers
116 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
70 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 ...
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.
2
votes
1answer
46 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 $\operatorname{tr.deg}_k \mbox{Frac}(k[x_1, \dots, x_n]/(p)) = n-1$?
1
vote
1answer
191 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 ...
4
votes
2answers
81 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 ...
5
votes
2answers
205 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 ...
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 $K$ ...
0
votes
1answer
36 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 ...
0
votes
1answer
33 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? If this is not true, what can i add to ...
3
votes
1answer
47 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
51 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 ...
6
votes
2answers
59 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?
1
vote
1answer
56 views

Conditions for a quotient module to be Noetherian

I'm solving this problem from "Introduction to Commutative Algebra" of Atiyah and Macdonald. Here is the problem: Let $M$ be an $A$-module and let $N_1, N_2$ be submodules of $M$. If $M/N_1, ...
1
vote
1answer
43 views

Can zero divisors be in the denominator when we localize rings?

Can we localize rings with zero divisors? Can those zero divisors be in the denominator? I thought defining $$\frac{a}{b}=\frac{c}{d} \text{iff }t(ad-bc)=0 \text{ where $b,d,t$ belong to the same ...
1
vote
1answer
37 views

If a proper ideal contains some power of a maximal ideal then the maximal ideal is the only prime ideal that contains the ideal.

Let $R$ be a commutative ring with $1$ and $\mathfrak{m}\subset R$ be a maximal ideal. Show that if $I\subset R$ is a proper ideal containing $\mathfrak{m}^n$ for some $n\geq 1$, then $\mathfrak{m}$ ...
1
vote
0answers
64 views

Reid, Undergraduate Commutative Algebra, Exercise 0.23

Let f $\in$ A; if f is reducible then the principal ideal (f) is contained in a bigger principal ideal (f$_1$). Consider the following conditions on a ring A. (a) A is a UFD; (b) every incresing ...
0
votes
0answers
48 views

Proof that $\mathbb{Z}[\sqrt{-5}]$ is integrally closed

There are demonstrations on the Internet saying that the polynomial $$\left(x-\frac{a}{c}-\frac{b}{d}\sqrt{-5}\right)\left(x-\frac{a}{c}+\frac{b}{d}\sqrt{-5}\right)$$ is monic if and only if ...
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.
1
vote
2answers
66 views

Can $R[[x]]$ contain constants?

Consider the ring $R[[x]]$ of formal power series $\sum_{n=0}^\infty a_nx^n$ with coefficients in $R$. I was wondering whether $R[[x]]$ contains elements of $R$ (polynomials of degree $0$). I'm ...
0
votes
0answers
120 views

Algebraic proof of Ehrhart's theorem

Let $P \subset \mathbb{R}^d$ be a $d$-dimensional polytope, where all vertices lie on integral coordinates, and let $L(P,n)$ denote the number of integral lattice points contained in the scaled ...
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 ...
3
votes
1answer
39 views

Rings of algebraic integers

A basic question on algebraic numbers. If $L/K$ is a finite extension of number fields with respective rings of integers $\mathcal O_L$ and $\mathcal O_K$ then is it true that $\mathcal O_L$ is ...
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
46 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
2
votes
2answers
81 views

Why is $ \mathrm{Frac} ( A / \mathfrak{p} ) = A_{\mathfrak{p}} / \mathfrak{p} A_{\mathfrak{p}} $? [duplicate]

$ A $ is a commutative ring, $ \mathfrak{p} \in \mathrm{Spec} A $, $ A_{\mathfrak{p}} = ( A \backslash \mathfrak{p} )^{-1} A $, $ \mathrm{Frac} ( A / \mathfrak{p} )$ is the field of fractions of $ A / ...
3
votes
0answers
52 views

$M_{\mathfrak{p}} \otimes_{R_{\mathfrak{p}}} N_{\mathfrak{p}} = 0$ implies $M_{\mathfrak{p}} = 0$ or $N_{\mathfrak{p}} = 0$ [duplicate]

Studying commutative algebra I've found this statement: If $M$ and $N$ are finitely generated $R$-modules, with $R$ a commutative ring, and $\mathfrak{p} \subset R $ is a prime ideal, then ...
1
vote
0answers
42 views

Finitely generated idempotent ideal must be generated by an idempotent [duplicate]

Let $A$ be a commutative but not necessarily unital ring. How can we show that a finitely-generated ideal $I$ of a ring $A$ satisfying $I=I^2$ is generated by an idempotent element?
5
votes
1answer
81 views

(Co)homology of free symmetric algebra

Let $V$ be a (co)chain complex, and let $Sym(V)$ be the free differential graded-commutative algebra generated by $V$. Definition and examples below in case you don't know what I mean. Question: ...
0
votes
4answers
74 views

Maximal ideal in the ring of polynomials over $\mathbb Z$

Let $\mathbb Z[x]$ the ring of polynomials with integers coefficients in one variable and $I =\langle 5,x^2 + 2\rangle$, how can I prove that $I$ is maximal ideal. I tried first see that $5$ and ...