# 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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### The characteristic of the quotient fields of $\mathbb{Z}[x_1,\ldots, x_n]$

Let $R_n = \mathbb{Z}[x_1,x_2, \ldots,x_n]$ be a $n$ variable polynomial ring. Given maximal ideal $I$ of $R_n,$,I want to find about the characteristic of the field $R_n/I.$ If $n=1$, it is well ...
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### Show that $I(\overline{Y_1 \setminus Y_2}) = I(Y_1):I(Y_2)$

Let $X$ be an affine variaty in a commutative ring $R$ and let $,Y_1,Y_2$ be subvarieties of $X$. Show that in the coordinate ring $A(X)=K[x_1,...,x_n]/I(X)$, where $K$ is an algebraically closed ...
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### Find a non-trivial element in the class group of $\mathbb{Q}( \sqrt{−5})$.

a. Find a non-trivial element in the class group of $\mathbb{Q}(\sqrt{−5})$. b. Show that the class group of $\mathbb{Q}( \sqrt{−5})$ has order two. For part a: I know that the class group is the ...
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### Can we determine maximal ideals of $\mathbb{Z}[x, y]$ by using $\mathbb{Z}[x] \subset \mathbb{Z}[x, y]$?

I am trying to prove that the a maximal ideal $m \subset \mathbb{Z}[x, y]$ is of the form $m = (p, f(x), g(x, y))$, where $p \in \mathbb{Z}$ is a prime number, $f(x) \in \mathbb{Z}[x]$ is a monic ...
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### Trouble in verifying a closed subset is an ideal.

I'm currently reading the paper The ideal structure of the Haagerup tensor product of $C^{\ast}$-algebras and having difficulty in verifying a statement given in last paragraph of Page $112$. Let $A$ ...
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### The spectrum of the polynomial ring is $\text{Spec}K[X]=\langle x-c\rangle$

let $K$ be a field, prove $\text{Spec}K[X]=\langle x-c\rangle$ , $c\in K$ The case where $\langle x-c\rangle\subset \text{Spec}K[X]$ is almost trivial if we use weak Nullstellensatz. We know that ...
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### Confusion about $\mathbb{Z}[x]$ being not a PID and generators of ideals of form $\left(f_1(x), \ldots, f_n(x)\right)$ [duplicate]

Not principal ideal. It's a well known fact, that $\mathbb{Z}[x]$ is not a PID, for example consider the following ideal \begin{align} I = \left(x, x + 2\right) = \{a(x)(x+2) + b(x)x| a(x), b(x) \in \...
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### Residue fields of $\mathbb R^\mathbb N$

Let $A:=\mathbb R^\mathbb N$ be the direct product of infinitely many copies of $\mathbb R$. Let $I_0$ be the ideal of all sequences of $A$ which have only finitely many nonzero entries. Then $I_0$ is ...
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### Show $f^{-1}(I)$ Ideal in $R$? [closed]

I'm trying to show that if $f: R \rightarrow S$ is a ring homomorphism, and $I \subset S$ is ideal, then $f^{-1}(I)$ is Ideal in $R$ I'm trying to show that is satisfies the conditions of an ideal, ...
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### minimal prime ideals of Noetherian ring

I’d like to show the following theorem. Theorem Let A be a ring. If A has infinite minimal prime ideals, then A isn’t noetherian. I tried as follows. Let $P_i (i \in N)$ denote minimal prime ideals. ...
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### "Cancelling" Prime Ideals

Several months ago I asked a question about Cancellation Law of Ideal Multiplication. I believe that the question was too vague and the setting was too general. Now I am interested in learning the ...
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### Equivalence of ideals of holomorophic functions seen as real smooth

Given two (radical, prime) ideals $I_1$ and $I_2$ of $\mathcal{O}_{\mathbb{C}^2,0}$ generated by holomorphic functions in two variables $g_1(z_1,z_2)$ and $g_2(z_1,z_2)$ (resp.), one can also consider ...
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### Show statements about "sums" and "square roots" of ideals

Let $R$ be a commutative ring with $1$ and $I,J$ ideals of $R$. Show: (1) $I+J:=\{i+j|i \in I,j \in J\}=\langle I \cup J \rangle_R$ (2) $\sqrt{I} := \{x \in R | x^n \in I, n\in \mathbb{N}\}$ is an ...
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### Extension and contraction of prime ideals by ring homomorphism

Let $A$ and $B$ be commutative rings with $1 \neq 0$. Let $\varphi$ be a ring homomorphism with $\varphi(1) = 1$. We consider the extension and contraction of $\varphi$. Let $P \subset A$ be a prime ...
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### The meet of two minimal generators of a stable ideal in a polynomial ring

Let $k$ be a field and let $R$ be the polynomial ring $k[x_1,\ldots,x_n]$. Let $I$ be a monomial ideal of $R$. We say that $I$ is stable if it satisfies the following "exchange property": ...
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### Maximal ideal $M$ is prime in commutative (non unital) ring A such that $A^2\neq0$.

Is this proof correct? $M$ is maximal $\iff A/M$ has no proper non trivial ideal $M$ is prime ideal $\iff A/M$ is a domain Now, suppose $A/M$ is a commutative ring with no proper non trivial ideal and ...
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### Find all left ideals of $M_n(\mathbb{C})$

I am taking an introductory abstract algebra class, closely following Gregory T. Lee's Abstract Algebra // An Introductory Course. I have been asked to find all left, right, and two-sided ideals of ...
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### What $R$-modules $M$ and ideals $I$ satisfy $IM=M$?

Let $R$ be a unital associative ring and $I\subseteq R$ be an ideal. Let $M$ be a (left) $R$-module. What unital associative rings $R$, $R$-modules $M$ and ideals $I\subseteq R$ satisfy $IM=M$? Do ...
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### Is $f\in \mathbb{C}[x]$ a non-unit if $\mathbb{C}[x]/(f)$ is a field? [duplicate]

I am trying to write a proof showing that, with $f\in \mathbb{C}[x]$ and $R=\mathbb{C}[x]/(f)$, $R$ being a field $\implies$ $f$ linear. I have written a rough version of my proof however going back ...
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### Let $I=(3, \sqrt{-14}-1)$ be an ideal in $\mathbb{Z}[\sqrt{-14}].$ Prove that $I, I^2, I^3$ are not principal but $I^4$ is.

For $I$ and $I^2$ I can directly calculate the product and then apply the norm trick to get a contradiction that if we assume they are principal, but I'm wondering that if there is any other good way ...
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### How to multiply these two prime ideals in Z[√(-5)]? [duplicate]

How to calculate the product of the below multiplication of prime ideals in Z[√(-5)]? (2, 1-√(-5))(3, 1+√(-5)) I know it can be firstly expressed as a non-principal ideal generated by three numbers in ...
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### Prove(or disprove) If $R/S$ is commutative ring then $R$ is commutative. [duplicate]
Given that $(R,+,\cdot)$ is a ring and $S$ is an ideal of $R$, then $R/S$ is a quotient ring. Is there any example such that $R/S$ is commutative ring but $R$ is not commutative ring ?
### Coming up with an alternative example to show that $IJ$ and $I \cap J$ can be different
Consider the following question where the ring is assumed to be commutative. For ideals $I, J$ of a ring $R$, their product $I J$ is defined as the ideal of $R$ generated by the elements of the form \$...