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

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Endomorphism ring as a set of matrices

Let $A=\mathbb Z[\sqrt{-5}]$, and let $I=(2,1+\sqrt{-5})$ (which is known to be a non-principal ideal of $A$ with $I^2=2A$). If we put $P=A \oplus I$, my question is: "why the endomorphism ring of ...
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judge if nilradical equals jacobson radical

judge if nilradical equals jacobson radical 1)a noetherian ring that is not a artin ring. 2)a local integral domain that is not a field. 3)a integral domain with only finite number of ...
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on the proof of Theorem 4.3.2 in Bruns & Herzog ``Cohen-Macaulay Rings" (Gotzmann's regularity theorem)

The theorem and the first part of its proof is shown below: In particular, the authors conclude (2 lines below equation (2)) that $(i): P_R(n) = {n + a_1 \choose a_1}+\cdots+ {n+a_r -(r-1) \choose ...
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Example of non noetherian ring and noetherian $\Bbb Z$-module

a non Noetherian ring that is a Noetherian $\Bbb Z$-module a Noetherian ring that is a non Noetherian $\Bbb Z$-module I have no idea in 1, and I'm not sure if $\mathbf{Q}$ is right for 2? ...
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Why field of fractions of $k[x_1,x_2,…]$ is Noetherian? [on hold]

the classical counterexample of a subring of a noetherian rings that is not noetherian is $k[x_1,x_2,...]$, which is not noetherian, but the field of fractions of $k[x_1,x_2,...]$ is, can anyone ...
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44 views

every ideal is contained in a maximal ideal

The statement is: In a commutative ring with 1, every proper ideal is contained in a maximal ideal. and we prove it using Zorn's lemma, that is, $I$ is an ideal, $P=\{I\subset A\mid A\text{ is ...
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Gröbner Basis and linear basis

Let $I$ be an ideal of a polynomial algebra $A$ with a Gröbner basis $G$. Suppose we know how to describe the leading terms of all elements in $G$, denoted by $\{i_1,\dots,i_k\}$, so that we can give ...
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Are the two ways of creating an $S^{-1}A$ algebra equivalent?

Let $f:A\to B$ be a ring homomorphism and $S$ be a multiplicative set, define $S^{-1}B$ to be $B\times S$ with equivalence relation $(b,s)\sim(b',s')$ iff $\exists t\in S$ such that $t(sb'-s'b) = 0$. ...
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41 views

MCS meet all prime ideals

let A be a commutative ring, is there any multiplicatively closed subset S (not containning 0), s.t. every prime ideal in A intercept S is not empty? My thinking is that there is 1-1 ...
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157 views

Where does the proof for commutative rings break down in the non-commutative ring when showing only two ideals implies the ring is a field?

We know in a commutative ring, if the only ideals are trivial and the whole ring, then the ring is a field, which is proved by every ideal is contained in a maximal ideal, which is proved by Zorn's ...
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maximal analytic spread

definition from Bruns-Herzog: It is easy to see that if $I$ is a $m$-primary ideal of $R$ then $ \lambda (I)= \dim R$. I wonder if the converse is true? if $ \lambda (I)= \dim R$, can one ...
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1answer
29 views

Proof of Steinitz Theorem

I want a source containing the proof of Steinitz Isomorphism Theorem stating: For any Dedekind domain $R$ and any two nonzero ideals $I$ and $J$ of $R$ we have $I⊕J≅R⊕IJ$. Thanks!
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Ramification group of valuations - need terminology

I am lost and need some terminology (also hopefully sources). Let $L/K$ be a Galois extension, and $w$ be a valuation of a $L$, lying above a valuation $v$ of $K$. Notice that I do not suppose that ...
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immersions and finite morphisms

I have the following question: Let $X \subset \mathbb A^n$ be an affine variety. Prove that the immersion $i\colon X \hookrightarrow \mathbb A^n$ is a finite morphism. I know that the ...
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37 views

Irreducible elements and unique factorization domain

Let $P=\{\frac{a}{3^n} : a \in \mathbb{Z}, n \in \mathbb{N}\}$. a) Which elements are irreducible in $P$: 4, 5, 6, 9, 10, 15? b) Find out, which one of rings: $ P$, $\mathbb{Z}[i\sqrt{5}]$, $P[x]$ ...
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67 views

Is ideal prime or maximal? [on hold]

Find, whether or not given ideal of $\mathbb{Z}[x]$ ring is prime or maximal and describe the quotient ring : a) $J_1 = (x-5)$ b) $J_2 = (3, x+5)$. How can I do that?
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If $P \in Supp(M)$ prove that $P$ contains a prime ideal $Q$ with $Q \in Ass_R(M)$.

My problem is below, Let $M$ be an $R$-module. The set of prime ideals $P$ of $R$ for which the localization $M_P$ is nonzero is called the support of $M$, denoted $Supp(M)$. The set of prime ideals ...
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Units of $\overline{\mathbb{Z}}$

What are the units of $ \overline{\mathbb{Z}} $ (the ring of algebraic integers)? I know all roots of monic polynomials with constant term 1 are units, but are there any others?
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Regular Local Ring

Let $Y$ be an affine variety in $\mathbb{A}^n_k$ and $\mathfrak{i}$ its corresponding ideal. We use the notation $A(Y) = k[x_1,...,x_n]/\mathfrak{i}$ for the coordinate ring of $Y$. Pick a point $p\in ...
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Categorical Interpretation of Localization

At the very beginning of Ravi Vakil's amazingly famously amazing and famous notes on algebraic geometry, he remarks that some familiarity with localization and prime ideals is useful. I don't know ...
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A question on Artinian and Noetherian rings.

All rings are commutative and unital. Suppose that $A$ is a ring in which the zero ideal can be written as a product of maximal ideals of $A$. I try to prove that $A$ is Noetherian if and only if ...
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Is this particular module flat?

Let $A=k[x^2,xy,y^2]\hookrightarrow B=k[x,y]$, where $k$ is a field. Is $B$ flat over $A$? I am guessing the answer is no. My first thought is, since $B$ is integral over $A$, so it's finitely ...
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Determinant bundle of a tensor product

Let $X$ be a ringed space (for example, a scheme or a manifold). If $V$ is a locally free $\mathcal{O}_X$-module of rank $n$, then $\mathrm{det}(V) := \Lambda^n V$ is a locally free ...
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The interpretation of ideals of a ring.

Ideals of a commutative ring (I have only studied the commutative case) are thought of as generalized numbers (in algebraic number theory) and as ring homomorphisms (through the ideal as kernel ...
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Commutative ring is sum of two ideals iff $x \to (x + I, x + J)$ is surjective.

I'm stuck on this exercise and any help would be well appreciated: Let $R$ be a commutative ring with ideals $I,J$. Show that $R=I+J$ if and only if $\phi(x)= (x + I, x + J)$ is surjective from ...
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71 views

Castelnuovo-Mumford regularity of Cohen-Macaulay modules

Let $S=K[X_1,\ldots,X_n]$ and $M$ be a Cohen-Macaulay $S$-module. This equality holds $$ \operatorname{reg}(M)=\dim(M)+\max\{i\in\mathbb{Z}\colon P_{M}(i)\neq H(M,i)\}. $$ It's been proved in ...
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30 views

Picard group of $\mathbb Z[\sqrt{-5}]$

I search for a simple proof for the fact that $\operatorname{Pic}(\mathbb Z[\sqrt{-5}])=\mathbb Z/2\mathbb Z$, where $\operatorname{Pic}(R)$ is the Picard group of the ring $R$ - the set of ...
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relation between units and non zero divisors in a ring

I can prove that in finite commutative ring, non zero divisors are units. My question is if the reverse also true. I mean, units are non zero divisors? And what about the commutative infinite rings?
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Can $ℂ$ be viewed as a (nontrivial) field of fractions?

Is there an interesting ring $S ⊂ ℂ$ such that $ℂ = Q(S)$? I’m thinking no, but how can I prove it?
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Proving that a certain local ring is regular

I understand that this is a special case of the Jacobian criterion, but I was hoping that there was a simpler argument to prove it than for the criterion itself (I don't fully understand the proof of ...
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1answer
50 views

Nakayama's lemma, second version

Let $R$ be a commutative ring with identity, $J$ an ideal that is contained in every maximal ideal of $R$, and $A$ is finitely generated $R-$ module. If $R/J\otimes _R A=0$, then $A=0$. ...
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Are finitely presentable modules closed under extensions?

If $0 \to A \to B \to C \to 0$ is an exact sequence of modules, and $A$ and $C$ are finitely presentable, then is $B$ finitely presentable? The answer is "yes" if we replace modules with groups, ...
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one to one correspondence of Ideals in a ring and its localization

Let $A$ be a commutative ring, and $S$ a mutiplicatively closed subset. In my text book, it is stated that: there is one to one correspondence of prime ideals in ring $A$ (not meeting $S$) and ...
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integral closure of a ring

I have (probably) a very simple exercise in commutative algebra. $k$ is a field and $A = k[x,y]/(x^3 - y^2)$. $\phi : A \to k[t]$ is a morphism defined by $\phi(x) = t^2, \phi(y) = t^3$. Show that ...
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Question about geometrical invariant

Assume $R$ is ring and $I $is ideal of $R $ The property of ideal $I$ was defined Geomerical properties which only depend on radical of $I$ For example varieties and projective varieties with ...
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Is Orzech's generalization of the surjective-endomorphism-is-injective theorem correct?

In math.stackexchange answer #239445, Makoto Kato quoted a statement from the paper Morris Orzech, Onto Endomorphisms are Isomorphisms, Amer. Math. Monthly 78 (1971), 357--362. The statement ...
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62 views

Describing $Spec(\mathcal{O}_K[X])$

Let $K$ be an algebraic number field and $\mathcal{O}_K$ its ring of integers. I am trying to describe $Spec(\mathcal{O}_K[X])$ in terms of fibers of the map $g: Spec(\mathcal{O}_K[X]) \rightarrow ...
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prove that this ideal is radical

Let $A=\mathbb k[x,y,z]$ and let the ideal $$ I=(z-1,x^2-y).$$ I need to find $rad(I)$ but i don't know how. I think that this ideal is radical but I don't know good criteria for doing that =(
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Finite Extension of Integral Domains.

Let $D\subset E$ (integral domains), with fraction fields $k\subset K $. Suppose that $E$ is integral over $D$, and $E$ is $D$-module finitely generated. My question is: $[K:k]$ is finite? Thank ...
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1answer
34 views

Ideals agreeing in a localization

I have an integral scheme $X$, and two coherent ideal sheaves $\mathcal I$ and $\mathcal J$ on $X$. I know there is a (maybe not closed) point $x$ of $X$ such that $\mathcal I$ and $\mathcal J$ ...
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The local rings of $xy=0$ and $xy+x^3+y^3=0$ are not isomorphic, but have isomorphic completions?

I know that if you have a commutative local ring $R$, and you take its completion $\widehat{R}$ the inverse limit of the $R/\mathfrak{m}^i$, you get another local ring. However, nonisomorphic local ...
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Flatness and Cohen-Macaulay rings

Let $A$ be a local Artin ring, $R$ a local Noetherian ring, $f:A \to R$ a flat morphism and $R$ is cohen-Macaulay. Let $I$ be an ideal in $R$ such that $R/I$ is also Cohen-Macaulay. Under what ...
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Proof of the Jacobian criterion - book of Eisenbud

I could really use some help understanding a statement in the last part of the proof of the Jacobian criterion in "Commutative Algebra with a view toward Algebraic Geometry" by D. Eisenbud, namely: ...
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Proving equivalent versions of faithfully flatness.

I was reading a proof of the the following theorem from Matsumura (p.47) There was something confusing about $(3) \implies (2)$ and $(2) \implies (1)$. Question 1 Here, it says $M \not= ...
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Is the localization of a ring $R$ at a prime ideal a finitely generated algebra over $R$?

Let $R$ be a ring and let $S=\{1,s,s^2,s^3,\dots\}$ be a multiplicative system of $R$. Consider the canonical map $R\rightarrow S^{-1}R$. Is $S^{-1}R$ a finitely generated algebra over $R$? It looks ...
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Atiyah-MacDonald Ch. 4 exercise 20: what's the module analogue of $\sqrt{\mathfrak{a}+\mathfrak{b}} = \sqrt{\sqrt{\mathfrak{a}}+\sqrt{\mathfrak{b}}}$?

Atiyah-MacDonald exercises 20-23 in chapter 4 develop a theory of primary decomposition for modules, in analogy with the theory developed in the chapter for rings. Exercise 20 begins with this ...
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Castelnuovo-Mumford regularity of a Veronese subring

I've faced a problem while reading a paper. It is mentioned to be trivial but I couldn't prove it. I'd appreciate if you can lead me to some resources or if you can prove it for me. Thank you. ...
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Are any of these rings isomorphic?

As part of my ongoing struggle to understand the complex conics, I've reached the following problem: Let $Q_1 = x^2 + y^2$, $Q_2 = x^2 - 1$, and $Q_3 = x^2$ be polynomials in $\mathbb{C}[x,y]$. ...
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1answer
45 views

Can a multiplicatively closed subset contain zero?

Let $A$ be a ring and $S$ be a multiplicatively closed subset. Can $S$ contain $0$? If so what will happen if we do $S^{-1}A$? A concrete and easy example coming to my mind is $A = \Bbb Z$, and $S = ...
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Classes of rings C[x,y]/(x²+cy²+ey+f) [duplicate]

I have a question. I would like to describe the classes of rings that appear in $\mathbb{C}[x,y]/I$ up to isomorphism, where $I=(Q)$, $Q=x²+cy²+ey+f$, $c,e,f\in\mathbb{C}$. $Q$ comes from ...