Questions tagged [ideals]

An ideal is a subset of ring such that it is possible to make a quotient ring with respect to this subset. This is the most frequent use of the name ideal, but it is used in other areas of mathematics too: ideals in set theory and order theory (which are closely related), ideals in semigroups, ideals in Lie algebras.

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Given 3 idelas who are coprime with eachother by pairs. Is the intersection between two of them coprime with the third?

Context Hello, what I'm currently trying to prove is the generalization to this equality $JK = J \cap K$ which is: $$\prod_{i=1}^{n}I_i=\bigcap_{i=1}^{n}I_i$$ I was told this could be done using ...
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$R\neq0$ com ring w/ 1.If $I$ is an ideal of $R$ such that $1+a$ is a unit in $R$ for all $a\in I$ then $I$ is contained in every maximal ideal of $R$

Question: Let $R$ be a nonzero commutative ring with $1$. If $I$ is an ideal of $R$ such that $1+a$ is a unit in $R$ for all $a\in I$ then $I$ is contained in every maximal ideal of $R$. My apporoach:...
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prove or disprove: if a domain has complete factorizations, the accp holds [duplicate]

Prove or disprove (with a counterexample) the following: if $R$ is an integral domain where every nonzero nonunit can be written as a product of irreducible elements, the ascending chain condition on ...
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showing a ring is a principal ideal domain

Suppose $R$ is a principal ideal domain. Let $S$ be a multiplicatively closed subset of $R$ not containing $0$. Show that $S^{-1}R$, the localization of $R$ by $S$, is a principal ideal domain. I ...
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Prove an element not belonging to the tight closure of an ideal

I'm working on a ring $R=\mathbb{F}_7[x,y,z]/(x^2+y^3+z^5)$ and want to prove $x\notin(y,z)^*$, the tight closure. First I want to find a test element, which can be obtained from the Jacobian ideal. ...
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If $I$ and $J$ are Ideals of a commutative ring $R$ where $I + J = R$, then $c \equiv a \pmod I$ and $c \equiv b \pmod J$.

Let $R$ be a commutative ring with unity. Let $I$ and $J$ be Ideals of $R$ and let $I + J = R$. I want to show that if $a,b \in R$, then there exists some $c \in R$ such that $c \equiv a \pmod I$ and $...
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Maximal elements of chain of colon ideal is prime ideal.

Let $A$ be commutative ring with $1$. Suppose $I$ is an ideal of $A$. Define $(I:x)=\{y\in A : yx\in I\}$. Then $(I:x)$ is an ideal and it is called colon ideal. Consider set of all ideals of such ...
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Is closure of a proper ideal in a unital Banach algebra also proper?

Let $A$ be a unital Banach algebra. Let $J$ be a proper ideal of $A.$ Can it be concluded that $\overline {J}$ is also a proper ideal of $A\ $? I know that closure of an ideal is also an ideal but I ...
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Integral ring extension (Problem with quotient rings)

Let $A$ and $B$ rings such that $A\subset B$ is an integral ring extension. $\textbf{Proposition.-}$ If $J$ is an ideal of $B$ and $J\cap A =I$, then $\dfrac{A}{I}\subset \dfrac{B}{J}$ is an integral ...
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Does $A/I \simeq A$ implies $I =0$?

Let $A$ be a unital algebra over a field, and $I$ an ideal of $A$. If there is an algebra isomorphism $A/I\simeq A$, does it implies that $I=0$? If no, then for what type of algebras can this be true? ...
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The structure of ideals in $C[0,1]$

Consider $R = C[0,1]$, the ring of real valued continuous functions on the interval $[0,1]$. Let $I$ be a proper ideal in $R$. Then there exists $S \subset [0,1]$ such that $I = I_S := \{f \in R : f(x)...
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Prove that $I$ is not a principal ideal

If $R$ is the ring that consist of all sequences of integers $(a_1,a_2,a_3,...)$, with $$(a_1,a_2,...)+(b_1,b_2,...)=(a_1+b_1,a_2+b_2,...)$$ and $$(a_1,a_2,...)(b_1,b_2,...)=(a_1b_1,a_2b_2,...)$$ And $...
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A question about the product of two ideals

In my algebra textbook appear the following: If $I$ and $J$ are ideals, then $I$ and $J$ are both subsets of $I+J$ . But by the absorption property, $IJ\subseteq I\cap J$. The first part is clear to ...
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Space of compact operators is the only proper closed two sided ideal of the space of all bounded operators.

Let $\mathcal H$ be a Hilbert space. Then $\mathcal K(\mathcal H)$ is the only proper closed two sided ideal of $\mathcal B(\mathcal H).$ I am following Rajendra Bhatia's notes on Functional Analysis....
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$\mathfrak{Jac}(A[x])=\mathfrak{Nil}(A[x])$ [duplicate]

Let $A$ be a commutative associative ring. Show that $\mathfrak{Nil}(A[x])=\mathfrak{Jac}(A[x])$. Since $\mathfrak{Nil}(A[x])=\bigcap\limits_{\mathfrak p\triangleleft A[x]}\mathfrak{p}$, it's ...
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Bijection between the ideals of $R$ contained in $\mathfrak{p}$ and $\text{Spec}(R_\mathfrak{p})$

This is part of 4.11 from Chapter V of Aluffi's Algebra. The point is to prove that $R_\mathfrak{p}$ is a local ring by showing that there is an inclusion-preserving bijection between the set of prime ...
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Ideal generated by maximal ideal and some other element of integral domain.

Let $R$ be an integral domain, $M$ a maximal ideal of $R[x]$, $P$ be a prime ideal of $R$, and $M \cap R=(0)$. Let $0\neq p\in P $. Then prove that $(M,p)=R[x]$. If $R$ is just a field it is done as ...
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Ideals generated by minimal elements in Bezout Unique Factorization Domains [closed]

Let $R$ be a Unique Factorization Domain, and also a Bezout Domain. Given an ideal of $R$, and an element $a \in I$ with a minimal number of irreducible factors, show that $I=(a)$. I have quite a lot ...
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showing a ring is isomorphic to a localization of $R$ at $S$

Let $\mathbb{K}$ be a field. The domain of $f = \frac{g}h\in \mathbb{K}(x), $ where $g,h\in \mathbb{K}[x]$ is defined to be the set $Dom(f) := \{c\in \mathbb{K} : h(c) \neq 0\}.$ Let $S(c) := \{f\in \...
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Integral domain and proper factors

Let $R$ be an integral domain and $a,b\in R$ where $a\mid b$ but not $b\mid a$. I have to show that this implies that $Rb \subsetneq Ra$. The question is related to this one. He however proved the ...
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Let $f: R \rightarrow R[X]$ be the natural map, and let $I$ be an ideal of $R$. Show that $I \in {\rm Spec}(R) \iff I^{e} \in {\rm Spec}(R[X])$.

This is exercise in Commutative Algebra: Let $R$ be a commutative ring and let $X$ be an indeterminate; use the extension and contraction notation of 2.41 in conjunction with the natural ring ...
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What if the gcd between two polynomials is 1

I have this exercise in $A=Z[x]/(x^2-2, 7)$ and I have to say if $A$ is a field or not. I know that I should show that the ideal generated by the gcd of the polynomial is maximal but i can't see how ...
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Definition and ideals of $\mathbb{Z}/n\mathbb{Z}$

First, I want to ask what elements are in the rings $\mathbb{Z}/n\mathbb{Z}$, the book I have defines the rings $R/I=\{a+I| a\in R\}$ where $a+I=\{x\in R| x-a \in I\}$ then proceed to give an example ...
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102 views

find all the maximal ideal of $\mathbb{Z} /p^n\mathbb{Z}?$

find all the maximal ideal of $\mathbb{Z} /p^n\mathbb{Z}?$ I know that $p\mathbb{Z}$ is a maximal ideal in $\mathbb{Z}$ whenever $p$ is prime Here $\mathbb{Z} /p^n\mathbb{Z}\cong \mathbb{Z}_{p^n}$ ...
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How do I proof Groebner basis existence in $R[x_1, \dots, x_n]$?

In An Introduction To Groebner Bases from Loustaunau, it says that: We further note that the Noetherian property of the ring R, and hence of the ring $R[x_1, \dots, x_n]$, yields to the following ...
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Explicit form of intersection of two ideals

Let $J=\langle y^2-x^4,y^2-y^3-2y^2x-yx^2+y^2x^2+2yx^3\rangle\subseteq \Bbb{C}[x,y]$. Find $\sqrt{J}$. I'm using nullstellensatz to find $\sqrt{J}$. Finding $I(V(J))$ I was able to deduce that $\sqrt{...
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How to prove : $6\mathbb Z /24\mathbb Z $ is an ideal in $\mathbb Z /24\mathbb Z $ , what are elements of $6\mathbb Z /24\mathbb Z $

Prove : $6\mathbb Z/24\mathbb Z$ is an ideal in $\mathbb Z/24\mathbb Z$. Is this ideal maximal? Is $\mathbb Z $ /24$\mathbb Z $ an integral domain? I don't know what are elements of $6\mathbb Z/24\...
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If $(a)+(b)$ is a principal ideal then $(a)\cap(b)$ is also a principal ideal. [duplicate]

Suppose $R$ be a commutative ring with $1$ and $a,b \in R$. I have to show if $(a)+(b)$ is a principal ideal then $(a)\cap(b)$ is also a principal ideal. Suppose $(a)+(b)=(d)$ for some $d \in R$. ...
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Radical of the sum of modules

Let $R$ be a commutative ring (not necessarily with a unit), $M$ be some $R$-module, and N a submodule of the module M. We define (according to Zariski and Samuel, 1958) the radical $\sqrt N$ of the ...
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$(AB:A) = B$ for every finitely generated $B$ does not imply that $A$ is a cancellation ideal

I am stuck at the following exercise: Let $R$ be a commutative ring and let $A$ and $B$ be ideals in $R$ with $(AB:A) = B$ for every finitely generated $B$. Show that this does not imply that $A$ is ...
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1+I invertible implies power intersection of I equals 0 [duplicate]

If $R$ noetherian ring, $I \subset R$ proper ideal such that every element of $1+I$ is invertible. Show that: $$\bigcap_{n>0} I^n = (0)$$ Idea: We know that $I \subset J(R)$, with $J(R)$ the ...
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How can we change the generator of an ideal, without changing the ideal itself?

I have following question: How can we change the generator of an ideal, without changing the ideal itself? Now, I think that we can sort of simplify it and for example, take only the elements which ...
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If $I$ is an invertible ideal, then $I=(I^{-1})^{-1}$.

I am working on the following exercise: Let $R$ be an integral domain and let $I$ be an invertible (fractional) ideal in $R$. Show that $I=(I^{-1})^{-1}$. Does it suffice to just refer to the ...
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If $I \cap J \cap K = IJK$ for proper ideals $I$, $J$, and $K$ (not containing each other) then does $I \cap J = IJ$?

Let $R$ be a commutative Noetherian ring. Then, I cannot construct a counterexample (or proof) for the question posed in the title. If $R$ is a polynomial ring and all ideals involved are monomial ...
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Minimal Ideals In LA Semigroup

Theorem. For each ideal $I$ of an LA-semigroup $S$, there exists a minimal prime ideal of $I$ in $S$. can any one show the above result for me. I, will be very thankfull.
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Is there a difference between the characteristic of $O_k/\mathfrak{p}$ and the ideal norm $||\mathfrak{p}||$?

Let $L$ be number field and $K$ a finite field extension. Let $O_K$ be the ring of integers of $K$, and $\mathfrak{p}$ a prime ideal of $O_K$. I have the definition of the ideal norm: $$ ||\mathfrak{p}...
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Weak Version of the Going Down Theorem, Eisenbud Exercise 10.7

I've been working on the following exercise from Eisenbud's book on Commutative Algebra. Exercise 10.7: Show that if $R$ is an integral domain contained in the local ring $(S,Q)$, then there is a ...
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Extension of a radical ideal

If $A$ is a commutative ring, $I \subset A$ an ideal and $f:A \rightarrow B$ a ring homomorphism, then the extension of $I$, $I^e = \langle f(a): a \in I \rangle$ does not commute with the radical, I ...
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Examine whether $(X^2+4,X)$ is maximal ideal of $\Bbb Z[X]$

We want to examine whether the ideal $I=(X^2+4,X)$ is a maximal ideal of $\Bbb Z[X]$, which as we know is not a PID. This result tells us precisely which are the maximal ideals of $\Bbb Z[X]$. Thus it ...
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Prove $(X,Y) \subset \mathbb{C}[X,Y]$ is not free as $\mathbb{C}[X,Y]$-module using two isomorphisms

Consider the ring $R = \mathbb{C}[X,Y]$ and let $I = (X,Y)\subset R$ be the ideal generated by $X$ and $Y$. In the exercise I have proven that $R/I \otimes_R I \cong R/I \oplus R/I$ and that $Q(R) \...
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Exercise 4.4 in Atiyah & Macdonald [duplicate]

I am working on Problem 4.4 in Atiyah & Macdonald, which asks the reader to show the following: In the polynomial ring $\mathbb{Z}[t]$, the ideal $m = (2,t)$ is maximal and the ideal $q = (4,t)$ ...
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Proving $\dim (R[[X]]) = \dim (R) + 1$ by using Krull's Principal Ideal Theorem, $R$ noetherian

I was able to prove "$\geq$" by showing that every prime ideal $p \subset R$ can be extended to $p' \subset R[[X]]$, with $p'$ being a prime ideal in $R[[X]]$. For a chain $p_1 \subsetneq ......
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If $R$ is a principal ideal domain, then every proper ideal of $R$ is contained in a maximal ideal of $R$.

Assume a proper ideal $I_{1}$ of $R$ is not contained in any maximal ideal of $R$. Then $I_{1}$ is not maximal since $I_{1}$ contains itself. Then there is an ideal $I_{2}$ such that $I_{1}\subset I_{...
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Is the module $\mathbb Q[x]$ over $\mathbb Z[x]$ a Dedekind module?

Following the definition of Dedekind modules by Naoum and Al-Alwan (1996), a torsion-free module $M$ over an integral domain $D$ (with its field of fractions $K$) is called a Dedekind if every non-...
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Is there a non-constant function $f$ such that $f(n) - 1 = 0$ for all $n \in \mathbb{N}$?

So I've been given rings $R = Map(\mathbb{R},\mathbb{R})$ and $S = (a_{n})_{n \geq 0}$ such that $a_{n} \in \mathbb{R}$ and the ring homomorphism $$ \phi: R \rightarrow S\\ f \rightarrow (f(n))_{n} $$ ...
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41 views

How to show that in $\mathbb{Z}[\sqrt{-3}]$ holds $(2,1+\sqrt{-3})(2,1+\sqrt{-3}) \subseteq (2)(2,1+\sqrt{-3})$

I am stuck at the following exercise: Show that in $\mathbb{Z}[\sqrt{-3}]$ holds $$(2,1+\sqrt{-3})(2,1+\sqrt{-3}) = (2)(2,1+\sqrt{-3}).$$ The direction $\supseteq$ is clear since $(2) \subset (2,1+\...
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51 views

Quotient ring of sequences for an ideal

I have the following here: Let $R$ be the ring of sequences $(a_n)_{n\geq0}$with entries $a_n \in \mathbb{R}$, and let $I=\{(a_n)_n\in R|a_n=0 \text{ for all } n<5\}.$ a) Show that $I$ is an ideal ...
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How do I go between generators and elements of an ideal in $𝔽_2[x_1,…,x_n]$?

Part 1: Elements of ideals defined by generators (answered) Consider the ideal $I=⟨f_1,f_2,f_3⟩⊂𝔽_2 [x_1,x_2,x_3,x_4 ]$. $f_1,f_2$ and $f_3$ are generators of the ideal $I$. $f_1=(1−x_1)x_2 \\ f_2=...
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Determining which of the following polynomials is a coset of $I$

I have the following question here: Let $I = \{h\cdot(x-1)| h \in \mathbb{Z}[x]\}$, an ideal of $\mathbb{Z}[x]$, and let $g = x^3+2x^2-x-3$. Which of the following polynomials $f_j$ are in the same ...
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exercise on radical ideals [duplicate]

I know that given a commutative ring with unit the sum of two radical ideal is not a radical ideal. I would want to know if for example in the ring of polynomial in $n$ variables with coefficient in ...

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