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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Expressing ideal sheaf of effective Cartier divisor as product of two ideal sheaves [duplicate]

If on a scheme, ideal sheaves $\mathscr{J}_1 \cdot \mathscr{J}_2 = \mathscr{I}$, where $\mathscr{I}$ corresponds to an effective Cartier divisor, is it necessary that $\mathscr{J}_1,\mathscr{J}_2$ ...
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$a$ is unit if and only if $a+I$ is unit.

Let R be a ring and I is ideal of it. We want to prove or disprove the statement. Let $a$ be a unit. Then there is $a'$ such that $aa'=1$. The quotient ring R/I implies, $(a+I)(a'+I)=aa'+I=1+I$ which ...
Arda Yonet's user avatar
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pure morphism and containment of ideals

Let $R$ be a commutative Noetherian ring. An $R$-linear map $f:M\to N$ of $R$-modules is called pure if $f\otimes X:M\otimes_R X \to N\otimes_R X$ is injective for every $R$-module $X$. Now let $f: R\...
strat's user avatar
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"Simple ideal" of a semigroup

I am working through Mario Petrich's Introduction to Semigroups. Lemma I.3.11 states: If I is a simple ideal of a semigroup S, then I is the kernel of S The problem is that he has not defined "...
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Localization: $(x) R_{\mathfrak{m}}=R_{\mathfrak{m}}=\mathfrak{a} R_{\mathfrak{m}}$ for $x\in \mathfrak{a}$ but $x\notin \mathfrak{m}$

I have a commutative ring $R.$ Let $\mathfrak{a}$ be a non-zero ideal in this ring and $\mathfrak{m}$ be a maximal ideal. Also, let $x$ be a non-zero element of $\mathfrak{a}$ such that $x\notin \...
Haldot's user avatar
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Let $f\in A$ and $p$ be the minimal prime containing $f$. If $\dim(A_p)=0$ then $f$ (and every element of $p$) is a zero divisor.

Let $A$ be a Noetherian ring. Let $f\in A$ be any element, and let $p\subset A$ be minimal among those primes containing $f$. If $\dim(A_p)=0$ then $f$ (and every element of $A$) is a zero-divisor. I ...
user13121312's user avatar
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Let $R$ be an integral domain. Prove $(a)=\{ra|r\in R\}$ with $(a)$ being the ideal generated by $a$ [closed]

i have the above task description. i tried to solve the problem but i do not even have an approach idea. how is the fact that $R$ is an integral domain important for the proof?
macman's user avatar
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Prove that $R / J$ is a field and determine $|R / J|$ with $J=(1-2 i) \subseteq \mathbb{Z}[i] .$

Exam question: In the ring of Gaussian integers $\mathbb{Z}[i]$, consider the ideal $J=(1-2 i) \subseteq \mathbb{Z}[i] .$ (a) Prove that $R / J$ is a field and determine $|R / J|$. (Check: The order $|...
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An injection from the set of normal subgroups to subsets of irreducible representations

Some things I understand to be true: (1) A finite dimensional $\mathrm{C}^*$-algebra $A$ is of the form $$A\cong \bigoplus_{j=1}^NM_{n_j}(\mathbb{C})\qquad (n_j\in \mathbb{N}).$$ (2) With respect to ...
JP McCarthy's user avatar
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Why is $M_2 (F)$ a direct sum of two minimal left ideals?

Let $F$ be a field and $M_2 (F)$ denotes the ring of all $2\times 2$ matrices with entries in $F$. I was wondering if someone could help me about this claim: Why is $M_2 (F)$ a direct sum of two ...
Mahtab's user avatar
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Making sense of closed sets of a Zariski topology.

Zariski topologies are defined via closed-set definition; with the closed sets being algebraic varieties $V(I)$ of ideals $I\subseteq R$ of commutative rings $R$... But what are these varieties, ...
Simón Flavio Ibañez's user avatar
2 votes
2 answers
186 views

An easier example of a non-PID where every finitely generated ideal is principal

Say that an integral domain $\mathcal{X}$ is an almost-PID if $\mathcal{X}$ is not a PID but every finitely generated ideal of $\mathcal{X}$ is principal. The question of whether almost-PIDs exist ...
Noah Schweber's user avatar
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Determining mapping cone of free resolution

I am reading the concept of mapping cone of a resolution. I need help with the following. Let $I_1=\langle x_1^2-x_2 x_4, x_1 x_2-x_3 x_4, x_1 x_3-x_4^2,x_2^2-x_1 x_3, x_2 x_3-x_1 x_4,x_3^2-x_2 x_4 \...
Raman's user avatar
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Computing the height of an ideal...?

I hope I'm not overbearing in this site. Yes, I'm still struggling. If you can, I have a question about primary decomposition that still needs help, you can find it in my page. Now I wanted to find ...
Goffredo Valenza's user avatar
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Confusion about a result in Pierce's book Associative Algebras.

I am confused about two Lemmas in the book Associative Algebras by R.Pierce (pages 43-44). In Lemma d, since each right ideal of a can be considered an algebra over $A$ then if $N$ is a minimal right ...
Adam's user avatar
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Why do we take finite sums when defining the product of ideals?

Let $I,J$ be ideals of ring $R$. The product of two ideals is defined by $\sum_{\text{finite}}ab : a\in I,b\in J$. My question is why do we take finite sums when defining the product of ideals? For ...
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Are solutions to polynomial equations subideals in a polynomial ring?

I'm wondering how can i make sense of a partial solution to a two-system of polynomial equations. Let $\{f,g\}\subseteq K[x,y]$ be polynomials in a polynomial ring over the field $K$. Given the ...
Simón Flavio Ibañez's user avatar
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1 answer
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Find all ideals of $\mathbb{Z}[\frac{1}{2}]$ [duplicate]

The ring $\mathbb{Z}[\frac{1}{2}]=\{\frac{u}{2^n}\}$ is the localization of $\mathbb{Z}$ in regards to $S=\{2^n:n \in \mathbb{N}\}$. Find all ideals of this ring. I'm not sure what is meant by this ...
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Set of co-images form left ideal of ring

Let $K$ be a field and $R=K^{2 \times 2}$. Let $U$ be a subspace of $\mathbb{Q}^2$ and $L=\{A \in R: coim(A) \subset U\}$. Then $L$ is a left ideal of $R$. For $L$ to be a left ideal the difference of ...
Magne Seier's user avatar
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Existence of ideal

Let $R$ be a ring, $I$ a finitely generated ideal of $R$ and $J$ an ideal of $R$ with $J \subset I$. Show that $R$ has an ideal $M$ so that $J \subset M \subset I$ and for every ideal $N$ with $M \...
Magne Seier's user avatar
3 votes
1 answer
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Proving $I(E(\Bbb C))=\left< y^2-x^3-ax-b\right>$

Let $E/\Bbb C : y^2=x^3+ax+b$ be an elliptic curve over $\Bbb C$. Let $$E(\Bbb C)=\{(x,y)\in\Bbb C^2:y^2=x^3+ax+b\}$$ be the affine variety generated by $E$. Finally, let $$I(E(\Bbb C))=\{f(x,y)\in\...
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Upper bound on smallest power of radical contained in the ideal

Suppose $J \subseteq \mathbb{C}[x_1,\ldots,x_n]$ is an ideal, and $I = \sqrt{J}$ is its radical. How large can the smallest integer $e$ for which $I^e \subseteq J$ be, specifically in terms of the ...
anamay's user avatar
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The polynomial ideal associated to a prime ideal is prime

Let $A$ be a ring, $I \subset A$ a prime ideal and $\Phi : A \rightarrow A[x]$ the canonical inclusion homomorphism. Is $I[x]= \{ \sum_{k=0}^n a_k x^k : n\in \Bbb{N}, (a_k)_{k=0}^n \in I^n \}$ also ...
Superdivinidad's user avatar
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Fact checking: are these inclusion relations regarding algebraic varieties of polynomial ideals correct?

I'm studying inclusion identities within polynomial ideals theory. More precisely, i'm interested in the correspondece of an ideal $I\subseteq \mathbb{F}[\vec{x}]$ and its associated affine variety $\...
Simón Flavio Ibañez's user avatar
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2 answers
73 views

Is this quotient still a ring? [closed]

Let's say we have $\mathbb{Z}_3[X]\mathbin{/}(7x^2 + 1)$, such that we take the quotient of the polynomial ring $\mathbb{Z}_3[X]$ with that specific ideal. Is this still a ring, and does it make sense ...
simogne 's user avatar
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1 answer
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Is every left ideal of $M_n (F)[x]$ principal?

Let $F$ be a field and $M_n (F)$ the ring of $n\times n$ matrices with entries in $F$ and $n>1$. I know that that $F[x]$ is a PID. Trivially, $M_n (F)$ is not commutative and not a field. My ...
Mahtab's user avatar
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Is any ideal of a Banach algebra closed?

This is the proof I have obtained to show that any ideal of a Banach algebra is closed: If $\cal B$ is a Banach algebra, then let $I$ be an ideal (bilateral). Let $\{y_n\}_{n\in{\mathbb N}}$ a ...
stkcpc's user avatar
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3 votes
1 answer
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If $A$ is a Jacobson ring, so is $A[X]$

I am studying Jacobson rings, using this file by Matthew Emerton and this source. I am trying to understand the proof of the following: Theorem. if $A$ is a Jacobson ring (that is, $\mathfrak p= \...
Robert's user avatar
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Is $(I(R:_{Q(R)} I))^n$ generated by $(fI)^n$ as $f$ varies over $(R:_{Q(R)} I)$?

Let $(R, \mathfrak m)$ be a Noetherian local domain of dimension $1$ which is not a UFD. Let $Q(R)$ be the fraction field of $R$. If $I\subsetneq \mathfrak m$ is a non-zero, non-principal ideal of $R$ ...
Alex's user avatar
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1 answer
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Center of an ideal of a $C^\ast$-algebra.

If $A$ is a $C^\ast$-algebra and $I$ is an ideal of $A$, then $Z(I)=Z(A)\cap I$? where Z(.) is the center. I have read in a paper https://doi.org/10.1093/imrn/rnaa133 that it is well known fact that $...
Anmol Paliwal's user avatar
4 votes
2 answers
157 views

Some very basic problems in understanding the definition of algebraic variety

I just started learning myself some basic algebraic geometry and I have some trouble doing these rather elementary exercises. I am basically misunderstanding something fundamental so it's causing me ...
Arbatus's user avatar
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Let V be a finite-dimensional vector space, $End_{K}(V)$ the ring of endomorphisms. [duplicate]

Let $V$ be a finite-dimensional vector space, $End_{K}(V)$ the ring of endomorphisms. Furthermore, let $I$ be an ideal of $End_{K}(V)$, $I \neq {0}$. I want to show that for all $u, v \in V, u,v \neq ...
Newbie1000's user avatar
2 votes
1 answer
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Calculate the ideal generated by a variety

I'm taking an algebraic geometry course and I'm trying to solve the next problem: Take the curve $V$ parametrized by $(t,t^3,t^4) \in \mathbb{R}^3$. Show that $V$ is an algebraic set and find $\mathbb{...
emilio j's user avatar
1 vote
2 answers
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Unique Prime Ideal in $\mathbb{Z}[\zeta_6]$ of Residual Characteristic 3

I’m trying to find the unique prime ideal in $\mathbb{Z}[\zeta_6]$ of residual characteristic 3, but I’m having difficulty finding it. I computed that $\mathbb{Z}[\zeta_6]\cong \mathbb{Z}[x]/(x^2-x+1)$...
Daichi's user avatar
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Is an ideal generated by irreductible elements of a ring a radical ideal?

I'm wondering if an ideal generated by irreductible elements of a ring is a radical ideal, I can't seem to prove it nor disprove it.
emilio j's user avatar
1 vote
1 answer
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$I=r(I)$ iff $I$ is an intersection of prime ideals

I am trying to solve this problem. Let $I$ be a non trivial ideal of a ring $A$, and let $r(I)$ be its radical. Prove that $I=r(I)$ iff $I$ is an intersection of prime ideals. I got the right to ...
albertvvila's user avatar
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example of a $B$-bimodule $X$ that is not an $A$-bimodule.

Suppose that $(A, ∥ · ∥_A)$ and $(B, ∥ · ∥_B)$ are two Banach algebras such that $B$ is an ideal in $A$ and $∥·∥_A ≤ ∥·∥_B$. We know that each Banach $A$-bimodule $X$ is also a Banach $B$-bimodule. ...
A. Friend's user avatar
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1 answer
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Artin's Algebra, maximal ideals of $\mathbb{C}[x, y]/(y^2+x^3-17)$

I am trying to solve the following problem from Artin's Algebra. Let $I$ be the principal ideal of $\mathbb{C}[x, y]$ generated by the polynomial $y^2 + x^3 - 17$. Which of the following sets ...
Abced Decba's user avatar
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If $I\subseteq P_1\cup P_2$ with $P_1, P_2$ prime ideals of $R$, then $I\subseteq P_1$ or $I\subseteq P_2$.

Please note that I am not looking for a solution to the question in the title as it has been asked before. I was trying to prove this famous result to a friend: If $I$ is an ideal of a ring $R$ and $...
IAAW's user avatar
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2 votes
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The equivalences between points in a locale in constructive mathematics

I am currently following the definitions of the book Frames and Locales and those in the articles Pointfree topology and constructive mathematics and Topo-logie. There are at least three ways of ...
Dylan Facio's user avatar
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Is the ring $\mathbb{R} [x]$ Noetherian?

I aim to assess the soundness of my approach. In an exercise, the question is posed regarding whether $\mathbb{R}[x]$ is Noetherian, and my response is negative. To support this, I employ the ...
Santa-claus 's user avatar
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1 answer
64 views

When $\mathbb Z_3[x] / \langle x^2+1\rangle$, why $(2x^2 + 2x + 1 + \langle x^2+1\rangle) = (2x + 2 + \langle x^2+1\rangle)$? [closed]

When $\mathbb Z_3[x]/\langle x^2+1\rangle$, why is $(2x^2 + 2x + 1 + \langle x^2+1\rangle) = (2x + 2 + \langle x^2+1\rangle)$? I know $\langle x^2+1\rangle$ has $2x^2+2$ but I'm confused by the fact ...
The Big guy's user avatar
-2 votes
1 answer
65 views

Prove that $I=(I\cap R_1)\oplus (I\cap R_2)\oplus \dots \oplus (I\cap R_n)$ [closed]

Let $R$ be a ring with identity, $R=R_1\oplus R_2 \oplus \dots \oplus R_n$, where $R_i \cap R_j =\emptyset$, $I$ is an ideal of $R$. Prove that $$I=(I\cap R_1)\oplus (I\cap R_2)\oplus \dots \oplus (I\...
Yue Yu's user avatar
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1 answer
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Why my proof of every maximal ideal being prime is incorrect

I tried to prove that for a commutative ring $R$ with identity, every maximal ideal $I$ is prime. I think my proof was sort of going in the right direction, but this MSE answer was what I now realize ...
user1181399's user avatar
1 vote
1 answer
42 views

Extended and Contracted Ideals in Ring of Fractions

In some lecture notes on commutative algebra I'm reading, one defines for a multiplicative subset $S\subseteq A$ the canonical map $f:A\to S^{-1}A$, $a\mapsto \tfrac{a}{1}$, then for an ideal $I$ of $...
mathemagician99's user avatar
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1 answer
65 views

Homomorphism between direct product of two rings and a domain

Let $R_1, R_2$ be rings with 1 and let $R$ be a domain. Let $f : R_1 \times R_2 \to R$ be a ring homomorphism. Prove that one of the following statements holds: (a) there exists a ring homomorphism $g ...
Zijian's user avatar
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1 vote
1 answer
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Fault in the proof that radical prime implies ideal is primary.

I have found examples regarding how the radical of an ideal being prime might not imply that the ideal itself is primary. However I am having trouble finding the error in the following proof- let $I$ ...
nkh99's user avatar
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-1 votes
2 answers
73 views

I'm confused by the last step when showing that $\Phi$ is well defined in the proof of 2nd isomorphism theorem for rings? [closed]

$S$ is a subring of $R$, $J$ is an ideal of $R$, and $\Phi: (S + J) \to S/(S \cap J)$ where $\Phi(a + b) = a + (S \cap J)$. We need to show that $\Phi$ is well defined. Suppose that $a_1 + b_1 = a_2 + ...
The Big guy's user avatar
2 votes
0 answers
54 views

Quotient module $\mathscr{O}[[X]] / \langle X^t , \pi^r \rangle $ is of finite-length

Let $\mathscr{O}$ be a discrete valuation ring with uniformizer $\pi$ and ring of formal power series $\mathscr{O} [[X]]$. I can understand untuitively "why" it is the case that the quotient ...
mathieu_matheux's user avatar
2 votes
1 answer
189 views

If the quotient of an ideal is principal, is the original ideal principal?

Let $R$ be a unity conmutative ring and $I \subset J \subset R$ ideals of $R$. Is it true that if $J / I$ is principal, then $J$ is principal? This question has came to me on other excercise in which, ...
Superdivinidad's user avatar

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