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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Intuitively, why $\operatorname{spec}(S^{-1}A) \cong \lbrace \mathfrak{p} \in \operatorname{spec} (A)| \mathfrak{p} \cap S = \emptyset \rbrace$?
If $A$ is a commutative ring, $S$ is a multiplicative subset of $A$, I’d like to understand intuitively why is there a bijection between $\operatorname{spec}(S^{-1}A)$ and $\lbrace \mathfrak{p} \in \...
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simplicial complexes and associated ideals
Simplicial Complexes have Stanley Reisner Ideals in Abstract Algebra. From what I understand, the ideal is generated by listing the minimal missing edges, faces, facets etc. I've drawn a few examples ...
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How is the reduced Groebner base of the ideal of an inconsistent polynomial system of equations?
Given a polynomial system ($f_1=0,....,f_s=0$) and the ideal $I$ <$f_1,....,f_s$>.
How is the reduced Groebner base of $I$ in the case that the system has no solutions?
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Would this be correct to prove that two ideals are equal? [closed]
Given that reduced Gröbner bases are unique for any given ideal and any monomial ordering, would it be correct to prove that two given ideals, $I_1$ and $I_2$, are equal following this process?
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Characterisation of non-closed ideals in $C[0,1]$
Let $C[0,1]$ be the $C^*$-algebra of continuous functions $[0,1]\to\mathbb C$.
I understand that the closed (two-sided, $*$-closed) ideals in $C[0,1]$ are in correspondence with the closed subspaces ...
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Show that $\mathcal{I}(\bigcup_iX_i)=\bigcap_i\mathcal{I}(X_i)$.
Let $\mathcal{I}$ be the vanishing ideal. I understand that $\mathcal{I}(A\cup B)=\mathcal{I}(A)\cap\mathcal{I}(B)$ where $A,B\subset \mathbb{A}^n$. For any collection of subsets $X_i$ of $\mathbb{A}^...
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Clarification on Exponents in Prime Factorization of Ideals in Dedekind Domains and Number Fields
Let $R$ be a Dedekind domain and $I$ a proper ideal. Then I know $I$ can be expressed uniquely as a finite product of prime ideals:
$$
I = \prod_{\mathfrak{p} \text{ prime}} \mathfrak{p}^{n_{\mathfrak{...
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Under what condition(s) $1) $ is true for a general integral domain $\mathcal{R}$?
Let $\mathcal{R}$ be a commutative ring with unity. Consider the ring $\frac{\mathcal{R}[x]}{{\langle f(x) g(x) \rangle}}$
$$\frac{\mathcal{R}[x]}{{\langle f(x) g(x) \rangle}}\cong \frac{\mathcal{R}[X]...
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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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Show that $(x^3-y^3)$ is a radical ideal of $\mathbb{F}_2[x,y]$.
Since $(x^3-y^3)\subset\sqrt{(x^3-y^3)}$, what we need to show is that $\sqrt{(x^3-y^3)}\subset(x^3-y^3)$. For any $i\in\sqrt{(x^3-y^3)}$，there exists $k\geq 1$，s.t. $i^k\in(x^3-y^3)$. Suppose that $i\...
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Preimage of a maximal ideal is maximal in free ring extensions
Suppose that $A, B$ are commutative rings with identity, and we
have an injective ring homomorphism $A \hookrightarrow B$.
I would like to know wether or not the following proposition is true:
If $B$ ...
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Irreducible components, dimension and degree of projective varieties
I have this problem given to me in my review session for my algebraic geometry final:
Describe the irreducible components and compute the degree and dimension of $V_p(x_0x_2-x_1^2, x_0x_3-x_1x_2)\...
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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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Primitive and modular ideals of $C^{\ast}$-algebras
Let $A$ be a $C^{\ast}$-algebra and $I$ be a closed subspace of $A$. Then $I$ is called $\textbf{modular}$ if $A/I$ is unital $C^{\ast}$-algebra and $I$ is called $\textbf{primitive}$ if $I=\...
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The complement of a filter is an ideal
In the book I'm reading, the author defines a filter like this:
Given a non-empty set $X$ a filter over $X$ is a set $f \subseteq P(X)$ such that:
(i) $f \not= \emptyset$
(ii) If $S_{1}, S_{2} \in f$ ...
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Intersection of $C^*$-algebras under addition
Let $\mathcal{B}\subset\mathcal{A}$ be an inclusion of unital $C^*$-algebras. Let $\mathcal{C}$ be an unital simple subalgebra of $\mathcal{A}$ which is under the image of a faithful conditional ...
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About ideals contained in the union of prime ideals
I am trying to solve the following exercise: Let $R$ be a commutative ring (here a ring is assumed to always have a multiplicative identy) and consider the following property for an ideal $I$:
(1) For ...
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Let $\psi:\mathbb Z[x] \rightarrow \mathbb R$ be the homomorphism defined by $\psi(p(x))=p(\sqrt3)$.
Let $\psi:\mathbb Z[x] \rightarrow \mathbb R$ be the homomorphism defined by $\psi(p(x))=p(\sqrt3)$.
a) Prove that the kernel of $\psi$ is a principal ideal.
b) Find the subring $S$ of $\mathbb R$ ...
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let $R$ be a ring with identity 1 and $\psi$ a non-trivial homomorphism of $R$ in an integer domain $D$. Show that $\psi(1)$ is the identity of $D$ [duplicate]
let $R$ be a ring with identity element 1 and $\psi$ a non-trivial homomorphism of $R$ in an integer domain $D$. Show that $\psi(1)$ is the identity element of $D$
My try: I assumed that there exists ...
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Show that $\mathbb{Q}[x] \backslash \langle x^2-3 \rangle$ is isomorphic to $\mathbb{Q}(\sqrt{3})$ [duplicate]
I want to proof that show that $\mathbb{Q}[x] \backslash \langle x^2-3 \rangle$ is isomorphic to $\mathbb{Q}(\sqrt{3})$ here is my attempt using the fundamental theorem of homomorphisms.
Let's define $...
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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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Quotient space and the need of being a closed left ideal
I have a question regarding a remark in my lecture. We defined the Nullspace $\mathcal{N}_{\phi}$ of positive form $\phi$ on a $C^*$-algebra $\mathcal{A}$ as $\mathcal{N}_{\phi}=\{A\in\mathcal{A}\,|\,\...
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Check if the ideal generated by the set is equal to the ring
Let $\langle S \rangle_R$ be the ideal of $R$ generated by the set $S$.
$1.R=\mathbb{Z},S=\{105,70,42,30\}$
$2.R=\mathbb{Z} \times \mathbb{Z},S=\{(4,3),(6,5)\}$
$3.R=\mathbb{Z}_{101}, S=\{[75]_{\equiv ...
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Intermediate prime ideal
I'm having some difficulties with this problem ($\mathbb{K}$ is algebraically closed).
Let $R\subseteq R'$ be an integral ring extension and let $R$ be a finitely generated algebra. Let $P_1\...
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Neukirch's Proof of $\dim_k (\mathfrak{O}/\mathfrak{pO}) = [L:K]$
The setup is as follows. Let $\mathfrak{o}$ be a Dedekind domain, $K$ its field of fractions, and $L/K$ a finite separable field extension with $n = [L:K]$. Furthermore, let $\mathfrak{O}$ denote the ...
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A generalization on bilateral ideals from metabelian to the general case.
Let $G = X \ltimes H$ be a finite group, semidirect product with normal divisor $H$. Let $K$ be a field over which irreducible modules are totally irreducible for $G$. Let the Norm be $N = \sum_{g \in ...
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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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Trying to construct closed prime ideal of a $C^{\ast}$-algebra
Let $\mathcal{A}$ be a $C^{\ast}$-algebra. A subset $S$ of $A$ is called multiplicatively closed if $ 0 \notin S$ and $ab \in S$ for all $a, b \in S$.
There exists a closed prime ideal $P$ of $\...
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Principal Ideal Theorem and geometric interpretation (Kempf 2.6.3)
I just studied Theorem 2.6.3 on Kempf's "Algebraic Varieties", which is a geometric version of the principal ideal theorem. It states:
Let g be a non-zero regular function on an irreducible ...
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Question about proof in commutative algebra
Hey I have a question about how to continue a proof in commutative algebra. Suppose you have a Ring $R$ and a maximal ideal $m$. I want to proof that $(mR_m)^n=m^nR_m$. One direction is obvious. For ...
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Simplified proof of Chevalley's theorem for $A\to k$. [duplicate]
I am looking for a simplier proof for the following special case of Chevalley's theorem:
Theorem. If $A\subset k$ is a subring of a field and the inclusion $i: A\to k$ is finitely presented, then $\{(...
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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 ?
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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 $...
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