# 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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### 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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### 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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### 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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### 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 ...
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 ...