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, ...

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About the discriminant ideal

Let $E/K$ be a separable field extension of degree $n$, let $A$ be a Dedekind Domain which quotient field is $K$, and let $B$ be the integral closure of $A$ in $E$. Then we have that the ideal ...
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11 views

Inverse limit of ideals equal to expected ideal of inverse limit?

Suppose we have a map $(A_n \to B_n)_{n \in \mathbb N}$ of inverse systems of unital rings and a system $\mathfrak a_n \lhd A_n$ of ideals, one sent into the next under the maps $A_n \to A_{n-1}$. ...
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1answer
24 views

Direct Summands of PI Rings as Right Ideals

Is any direct summand of a PI-ring (polynomial identity ring) necessarily idempotent as a right ideal? The answer is yes for a special case of PI-rings, namely any direct summand of a ...
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1answer
45 views

Finding the maximal ideals of the quotient of a polynomial ring by an ideal

Let the field $k$ be algebraically closed, let $f(X) \in k[X]$ be a separable polynomial of degree at least $2$, let $$ B = \frac{k[Y,X]}{(Y^2 - f(X))} $$ and write $y,x$ for the images in $B$ of $Y$ ...
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1answer
58 views

Problems with proof of Krull's height theorem

I want to understand the proof of next Theorem. Let $A$ a Noetherian ring and $\mathfrak a=(a_1,...,a_n)$ a proper ideal of $A$. Let $\mathfrak p\in\mathrm{Spec}(A)$ a minimal ideal over ...
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1answer
20 views

Is a ring R, modulo an ideal I (generated by x), then modulo an ideal J (generated by n) the same as R modulo the ideal generated by (n,x)?

Is the following statement true? $$ R/(x,n) = \left[ R/(x) \right] / (n) $$ My thinking behind it was as follows: \begin{array}{ccc} \left[ R/(x) \right] / (n) & = & \{ r+(n) : r \in R/(x) ...
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1answer
41 views

Let $A$ a Noetherian ring and $ q \in\mathrm{Spec} (A)$. Then $q^{(n)} A_q=q^{n}A_q$.

Let $A$ be a Noetherian ring and $ q \in\mathrm{Spec} (A)$. Then $q^{(n)} A_q=q^{n}A_q$, where $q^{(n)}= \lbrace a \in A \mid \exists d \in A \setminus q\text{ such that }da \in q^n \rbrace$ and ...
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1answer
29 views

Can we find ( characterize ) all non-zero commutative Artinian rings $R$ for which $-1 , 1$ are the only units of $R$ ?

Can we find ( characterize ) all non-zero commutative Artinian rings $R$ for which $-1 , 1$ are the only units of $R$ ?
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1answer
30 views

Definition of primary ideal [duplicate]

I am confused with the definition of a primary ideal. The definition states that if $R$ is a commutative ring then $I$ is called a primary ideal of $R$ is the following condition holds. If $xy\in I$ ...
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1answer
53 views

The ideals $\langle y-x-1\rangle$ and $\langle x-2,y-3\rangle$ in $\mathbb C[x,y]$ are prime

The following is a quote from Wolfram MathWorld article about prime ideals. A maximal ideal is always a prime ideal, but some prime ideals are not maximal. In the integers, $\{0\}$ is a prime ...
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1answer
23 views

Sum of nil right ideals as an ideal

I have two questions: 1) If $S$ is the sum of all right nil ideals of a ring $R$ (with unity), is it true that $S$ is a two-sided ideal? It is clear for me that $S$ is a right ideal (and it is nil if ...
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1answer
38 views

Ideal quotient is principal

Let $R$ be a commutative ring with $1$ and $I$ an ideal. Also let $B$ be a principal ideal, and $A=\{a\in R\;|\; aB\subseteq I\}$. I want to show that if $A$ is also principal then $I$ is principal. ...
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30 views

Show that the union of a chain of ideals is an ideal.

Here is my proof: Let $I=I_1\cup\ I_2\cup\ I_3 \cup\ ..... \cup\ I_n$, $a\in I$ and $r\in R$. Then $a\in I_i$ for some $i$ varying from $1$ to $n$. Since $I_i$ is an ideal of $R$, we have $ar\in ...
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1answer
21 views

$N(R)$ when $R$ is a P.I. ring

The set of nilpotent elements $N(R)$ of a ring $R$ with identity is not necessarily a right ideal (or even a subgroup) as it is seen in the ring of $2×2$ matrices over $\mathbb Z$. But, my question ...
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48 views

Principal Ideal using coordinates?

I thought I understood principal ideals but now im stuck... I want to find the elements of the principal ideal $\langle(1,0)\rangle$ in the ring $\mathbb Z_3\times \mathbb Z_3$ with $+_3$ and $*_3$ in ...
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3answers
27 views

What are the ideals of $F_2[x]/\langle x^2 + x +1\rangle$? [closed]

Is it just the divisors of $x^2 +x+1$ in mod $2$ ?
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1answer
37 views

Show that some monomial ideal is primary

Show that $I=(X_{k_1}^{a_1},...,X_{k_s}^{a_s})$ is $(X_{k_1},...,X_{k_s})$-primary. I noticed that ...
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0answers
14 views

Show that $(X_{k_1}^{a_1},…,X_{k_s}^{a_s})$ is $(X_{k_1},…,X_{k_s})$-primary [duplicate]

Show that $I=(X_{k_1}^{a_1},...,X_{k_s}^{a_s})$ is $(X_{k_1},...,X_{k_s})$-primary, where $I$ is the ideal generated by the monomials $X_{k_1}^{a_1},...,X_{k_s}^{a_s}$ . I noticed that ...
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53 views

When a prime ideal is maximal differential ideal in a UFD?

Is the prime ideal $\langle X^{2}+Y^{2}-1\rangle$ a maximal differential ideal in differential ring $\mathbb{Q}[X,Y]$ with derivatives $D(X)=Y, D(Y)= -X$? I know there are maximal ideals like ...
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2answers
39 views

What are the principal ideals of $\mathbb Z$ [closed]

What are the principal ideals of $\mathbb Z$ I thought the answer would be $\mathbb Z$ and $\{0\}$. However, the answer says: $m\Bbb Z \subseteq\Bbb Z $ for $m > 0$ Can somebody explain why? ...
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1answer
18 views

Invariant factors and the elementary divisors of the group $(\mathbb{Z}/77 \mathbb{Z})^{\times}$

If $A$ is a ring with unit element $1 \ne 0$ let $A^{\times}=\{a \in A: a$ invertible$\}$. Find the invariant factors and the elementary divisors of the group $(\mathbb{Z}/77 \mathbb{Z})^{\times}$. ...
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17 views

Nilpotent element in $\mathbb{Z}/12\mathbb{Z}$ - Ideal

Let $A$ a commutative ring with the unit element $1 \not= 0$. $a \in A$ is a nilpotent element if there exists $n \in \mathbb{N}$ such that $a^n=0$. I have already prove that the set of nilpotent ...
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2answers
129 views

How can we find the prime ideals of $\mathbb{Z}_{12}$? [closed]

I have found that the maximal ideals of the ring $\mathbb{Z}_{12}$ are $(2)$ and $(3)$. Is this correct? How can we find the prime ideals of $\mathbb{Z}_{12}$ ?
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1answer
39 views

Product of two principal ideals in $\Bbb Z[x]$

I'm looking for an easy argument for the following question: True or false, and why: The product of two principal ideals in $\Bbb Z[x]$ is a principal ideal. I know that $\Bbb Z[x]$ is not a ...
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2answers
26 views

Generating system for $R$-module $M$ still generating $\mathfrak aM$?

Let $M$ be a module over a commutative ring $R$ with unity and $\mathfrak a$ an ideal in $R$. Let $m_1, ..., m_n \in M$ be a generating system for $M$ and $z \in\mathfrak a M$. Then there are $a_1, ...
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27 views

What is the correspondence between primary decomposition and algebraic geometry? [duplicate]

What is the correspondence between primary decomposition and algebraic geometry?
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1answer
39 views

$A\subset B$ if $A\cdot R[X] \subset B \cdot R[X]$? [closed]

Can we conclude $A\subset B$ if $A\cdot R[X] \subset B \cdot R[X]$ for ideals $A,B$ in $R$, where R is a commutative ring with unity and $A \cdot R[X]$ the ideal generated by the products $af$, ...
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1answer
16 views

Showing the sum of a C* subalgebra and ideal is itself a C* subalgebra

In my functional analysis class I was recently met with this in the context of C* algebras: Let A be a C*-Algebra and B is a C*-subalgebra of A and I an ideal of A. We are asked to show that $ B+I ...
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1answer
74 views

Exercise on radical ideal and formal derivatives

I need some help for solving the following exercise, because at the moment I'm a little bit lost and don't know where to start. Given a field $k$ with $\mathrm{char}(k)=0$ and a polynomial $f\in ...
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2answers
31 views

Reduced ring are SI?

A ring $R$ is called an SI-ring if for any $a\in R$ the right annihilator of $a$ is an ideal of $R$. It is equivalent to the following statement: "if $ab=0$ for $a,b\in R$ then $aRb=0$". Is it true ...
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1answer
33 views

Listing all the ideals of a quotient ring [closed]

I have no idea how to answer this question. Let R be the quotient ring $\mathbb Q[X]/(X^3 + X^2 + X + 1)$. How to list all the ideals of R? And how to determine whether each ideal is prime, ...
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42 views

Show that there are finitely many different principal ideals [duplicate]

Let $R$ be a U.F.D. and $0\neq d\in R$. I want to show that there are finitely many different principal ideals that contain the ideal $(d)$. $$$$ We have that $R$ is a U.F.D. iff $\forall r\in ...
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1answer
52 views

Show $Z(yf-1)$ is irreducible.

Question: $k$ is an algebraically closed field. Let $f \in k[x_1, \ldots, x_n]$ be an irreducible polynomial. Show that $Z(yf-1)\subseteq \textbf{A}^{n+1}$, with coordinates $x_1, \ldots, x_n, y$, ...
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3answers
66 views

Each prime ideal contains an idempotent element

An element of ring $e$ is called idempotent iff $e^2=e$. Let $R$ be a commutative ring that contains the identity element and a non-trivial idempotent element. I want to show that each of ...
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2answers
34 views

(0) is Maximal Ideal of Q(i) [closed]

Prove or disprove Ideal generated by $(0)$ is maximal ideal in $\mathbb{Q}(i)$.
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1answer
49 views

Can every polynomial generate an ideal?

Suppose an arbitrary polynomial $f$ in a polynomial ring $R$. Is $\langle f\rangle$ always an ideal? Helper parts Consider a finite polynomial ring. Let $R=R[x_1,\ldots,x_n]$. Is the answer ...
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0answers
30 views

How is “Binomial” defined in Algebraic Geometry?

I am learning ideal arithmetics and I was flabbergasted that $\langle x\rangle$ is binomial ideal, as observed with Macaulay2 here. $x$ is clearly not a polynomial with two terms. Then I read paper ...
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28 views

Binomial ideals that are not toric i.e. binomial and not prime?

Let $R$ be a ring. Toric ideal $I$ is binomial (generated by a binomial) and prime (the quetient ring $R/I$ is integral domain). Paper and corollary 1.3 is to determine whether an ideal is binomial. ...
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1answer
21 views

$R$ commutative with identity, $I$ finitely generated ideal. For each $r\in I$ there exists $m$ such that $r^mA\subseteq C$. Show $I^nA\subseteq C$.

I'm stuck on an exercise (E.VIII.4.1) from Algebra by Hungerford. It is stated as follows (sorry for the bad title; I didn't know how to fit it): Let $R$ be a commutative ring with identity and ...
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1answer
15 views

Is the difference of ideals an ideal?

I am studying ideals and noticed that $I+J$ is an ideal as noted here. However the paper does not discuss $I-J$ so: Is the difference of ideals an ideal?
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1answer
54 views

Has toric ideal something to do with torus?

I am studying ideals such as toric ideals but I am unable to find a consistent definition, it seems to be very general so please explain the origin of "toric ideal". Is there a geometric ...
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1answer
37 views

Each element is invertible [closed]

Let $R$ be a ring and let $I\subseteq R$ the only maximal right ideal of $R$. I want to show that each element $a\in R-I$ is invertible. $I$ is also an ideal. Could you give me some hints ...
2
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1answer
52 views

Algebraic number theory, Marcus, Chapter 3, Question 9

Question 9 in Marcus book. Let $K$ and $L$ be the number field such that $K\subset L$ and let $R,S$ be their algebraic integers, respectively. a) Let $I$ and $J$ be ideals in $R$, and suppose ...
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2answers
70 views

How can we show that $I$ is an ideal? [duplicate]

Let $R$ be a ring and $I$ the set of non-invertible elements of $R$. If $(I,+)$ is an additive subgroup of $(R,+)$, then show that $I$ is an ideal of $R$ and so $R$ is local. $$$$ I have done ...
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3answers
36 views

Norm of element $\alpha$ equal to absolute norm of principal ideal $(\alpha)$

Let $K$ be a number field, $A$ its ring of integers, $N_{K / \mathbf{Q}}$ the usual field norm, and $N$ the absolute norm of the ideals in $A$. In some textbooks on algebraic number theory I have ...
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1answer
76 views

Show that $I$ is an ideal

Let $R$ be a ring and $I\subseteq R$ the only maximal right ideal of $R$. I want to show that $I$ is an ideal. To show that $I$ is an ideal, we have to show that $I$ is a left ideal, right? ...
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0answers
32 views

$(2, 1+\sqrt{-5})$ is a prime ideal in $\mathbb{Z}[\sqrt{-5}]$ [duplicate]

I need a hint (just a hint please, not a full answer) to proving that $(2, 1+\sqrt{-5})$ is a prime ideal in $\mathbb{Z}[\sqrt{-5}]$. I'm trying to prove it via definition of a prime ideal and ...
1
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1answer
37 views

Show that $I^{(n)}$ is a primary ideal belonging to $P$ [duplicate]

Let $A$ be a commutative ring with identity, $P$ a proper prime ideal in $A$, $I$ a primary ideal belonging to $P$ and $n$ a positive integer. The ideal $(I^{n})^{ec}$ (extension and contraction being ...
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1answer
27 views

Let $A$ be a ring and $m_1,…,m_k$ maximal ideals

Let $A$ be a ring and $m_1,...,m_k$ maximal ideals of $A$, not necessarily different, and $F_i=m_1\cdots m_{i-1}/m_1\cdots m_i$. Because $m_iF_i=0$, $F_i$ can be made into a $A/m_i$-module defining ...
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0answers
37 views

Proof of commutative Artinian ring is Noetherian

I think that I have a proof, but it seems much simpler than all proofs that I can find on the internet. Hence I suppose that there must be a mistake in my proof. The commutative ring $R$ is ...