Mathematics Stack Exchange
Mathematics Stack Exchange

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.

An ideal $I$ in a ring $(R,+,\cdot)$ a subset $I\subseteq R$ such that $(I,+)$ is a subgroup of the additive group $(R,+)$ and $r\cdot x,x\cdot r\in I$ whenever $r\in R$ and $x\in I$ (i.e., $I$ is closed under multiplication by arbitrary elements).

This is the most frequent use of the name ideal, but it is used in other areas of mathematics too:

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A question about ideals.

Let $A$ and $B$ be arbitrary subsets of a ring. Then $V(A\cup B)=V(A)\cap V(B)$. Here, $V(X)$ is the set of prime ideals containing $X$. Let $W(X)$ be the set of ideals (any sort of ideals) containing the set $X$. Is $$W(A\cup B)=W(A)\cap W(B)$$…
freebird
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Decide belonging of an element to an ideal

Let $f_1=x^2-y, f_2=xy-z,f_4=xz-y^2$ be polynomials with coefficients in some field $k$. I want to prove that $f_2\notin (f_1,f_4)$. My attempt: by contraddiction, let $f_2\in (f_1,f_4)$. Then there exist $\alpha,\beta\in k[x,y,z]$ such that…
Danae Kissinger
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When does $IJ=(I+J)(I\cap J)$

It's clear that $IJ\supset(I+J)(I\cap J)$, but when is the reverse inclusion true? So far, the simplest counterexample I could find was $I=(x), J=(x^2,y)$ in $k[x,y]$.
pre-kidney
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problem concerning ideal of Z[X]

let $a$ a complex number , and $f$ be an irreducible polynomial with integer coefficients such that : $ f(a)=0$ 1) Show that the set : $\{ g(a) \mid g \in \mathbf{Z}[X]\}$ is a ring isomorphic to $\mathbf{Z}^{n}$ respect to their group…
mathfan
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If an ideal $I$ in a unital commutative ring equals the intersection of some prime ideals, then it equals the intersection of all prime ideals.

Let $I \subset R$ be an ideal of a unital commutative ring $R$. I have in my notes that TFAE: $I$ equals the intersection of some prime ideals containing $I$. $I$ equals the intersection of all prime ideals containing $I$. Now, I think the…
Ben123
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If $g \in I$ for an ideal $I$ over a commutative ring $R$, and $f \not \in I$, what can we say about $f-g$?

This is a really basic question: Let $R$ be a commutative ring and $I \trianglelefteq R$ be an ideal. Let $g \in I$ and let $f \not \in I$. Can we say for certain that $f - g \not \in I$? A classmate points out that you could say that $$g+I =I$$ so…
Ben123
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Why is this generated ideal really an ideal?

I have the following problem: Let $S\subset R$, where $R$ is a ring. We write $(S)$ for the smallest two sided ideal of $R$ containing $S$, i.e. $$(S)=\left\{\sum_{i=1}^n a_i s_i b_i: a_i, b_i \in R, s_i \in S, n\geq 0\right\}$$ So I mean that's…
user1294729
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Intersection of $a+b \mathbb{Z}$ and $c + d \mathbb{Z}$

Let $a $ and $ b $ be integers and consider the following subset of $ \mathbb{Z}. $ $$ a+b\mathbb{Z} = \{ a+bz \mid z \in \mathbb{Z} \}. $$ What is the intersection of $ a+b \mathbb{Z} $ and $ c+d \mathbb{Z} $ for integers $ a,b,c,d. $ Justify your…
pseudogeek
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Ideals in $\mathbb{Z}_n$

For which $n > 1$, the set of all not invertible elements of the ring $\mathbb{Z}_n$ is an ideal? This is where I got to: The ideal, surely, should consists only of zero- divisors (because each not invertible element is a zero- divisor). Also, all…
Simon Jachson
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Coprime ideals in ring

Given the commutative ring R with unity and $a,b,c \in R$, prove that if the following are coprime principal ideals $$ ⟨a⟩=\{\ ar\ |\ r∈R\ \}\ and\ ⟨b⟩=\{\ br\ |\ r∈R\ \} $$ and $a\ |\ bc$ (a divides bc), then $a\ |\ c$ (a divides c). I know that…
Paul
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Can you always define the sum of ideals?

Let $R$ be a ring and $I_1, I_2$ be ideals. Then the sum of these ideals is defined as: $I_1+I_2=\{s_1+s_2| s_1\in I_1\,\text{and}\, s_2\in I_2\}$ Is it always possible to define this sum of ideals, or just for a finite amount of ideals? Let…
Cornman
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Let R={f:{1,2,3,4,5,6,7,8,9,10}→Z2}Then which of the following are correct?

Let R={f:{1,2,3,4,5,6,7,8,9,10}→Z2} be the set of all Z2-valued functions on the set [1,2,3,4,5,6,7,8,9,10} of the first ten positive numbers.Then R is a commutative ring with point-wise addition and multiplication.Then which of the following are…
Halima.Khatun
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Radical İdeal-Nullstellensatz

$f=y{^2}+2xy-1$ and $g=x^2+1$.Prove $$ is not radical ideal. Hint :What is $f+g$ $?$ I have an idea: neither $x$ nor $y$ belong the ideal.but $(x\cdot y)^{2}\in I$.Since $x^{2}\in I$ ; thus $x\cdot y\in\sqrt{I}$.So $x+y\in\sqrt{I}$
Burak Kılıç
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The set of all continuous functions with compact support is ideal??????

AS far as ideal of a ring is concerned, it is not ideal. I am giving an counter example. f(x) = 1 for all x belongs to R. which is a continuous function with compact support. g(x) = x for all x belongs to closed interval 0 to infinity. …
anonymous
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An Example $\sqrt{(f^2,g^3)}\neq(f,g)$

Can you give an example for $f,g\in\mathbb K[x,y]$ and $\sqrt{(f^2,g^3)}\neq(f,g)$ where $(.)$ means the ideal In general how do you perform computations with ideals, if $f$ and $g$ were monomials then the equality should hold,…
user257
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