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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Regarding taking powers of prime ideals in a ring

My question is simple to ask: given some prime ideal $P$ in a ring $R$, we can talk about $P^2, P^3$ etc. but can we discuss $P^0$? Is there a convention that says $P^0 = R$, or is there something ...
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Filling in Proof: Well-definedness of depth(I,M).

From Eisenbud's Commutative Algebra with A View Toward Algebraic Geometry (Theorem 17.4): Let $M$ be a finitely generated $R$-module, where $R$ is Noetherian. If $$r= \min \{i : H^i(M\otimes K(x_1,...
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Prime and maximal ideal

If I have to show that ideal A is not maximal, is it enough to show that A is not prime because it is usually easier? Every maximal ideal is prime so if we have ideal that is not prime, it can not be ...
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Some ideal property in a local ring

If we change the ideal $$(X_1,X_2^2-X_1,...,X^2_{n+1}-X_n,...)$$ to $$(X_1^2,X_2^2-X_1,...,X^2_{n+1}-X_n,...)$$ in this problem, what is the answer to the raised question? Again, the new local ring ...
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Ideal generated by given integers verification.

The question reads: Find the positive generator of the smallest ideal in $\mathbf Z$ containing the following ideals: a. $(4)$ and $(18)$. My answer is $(m)=(4)$. b. $(6)$ and $(35)$. My ...
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If I and J are isomorphic ideals of a ring R, does it follow that $R/I \simeq R/J$?

The title pretty much sums it up. We know that $R/I \simeq R/J$ does not necessarily imply $ I \simeq J$. But does the converse hold? I can't find any counterexample and all my efforts in proving it ...
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38 views

Show that $(a) + (b)= R$ for $\gcd(a,b) = 1$

The question I am trying to solve it: Let $R$ be a principal ideal domain, $a,b\in R$. Suppose $\gcd(a,b) = 1$. Show that $(a)+(b)=R$. First I have tried to show that $(a)+(b)$ is in R: $\gcd(...
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Upper Nilradical of a Ring

If we define the upper nilradical of a ring as the sum of all nil ideals of the ring, how could we deduce from just this definition that this is a nil ideal? Thanks!
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Providing non trivial module morphisms

I'm starting to study modules, and I would like to get some counterexamples to naive ideas one has in the first approach to the subject. Does there exist an ideal I in A and a morphism $f:I \...
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38 views

Having difficulties showing this map is well defined?

Say $J$ is an ideal of $K[Y_1,...,Y_m]$ and $I$ of $K[X_1,...,X_n]$. Then $\phi:K[Y_1,...,Y_m]/J \to K[X_1,...,X_n]/I$ defined by $Y_i \mapsto f_i, i\in\{1, \dots ,m\}$ is well defined $\iff$ $J=(g_1, ...
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$k[x,y]/(xy-1)$ isomorphic to $k[x,\frac{1}{x}]$ [duplicate]

I mean clearly one uses the isomorphism $\phi$ that sends to $x$ to $x$ and $y$ to $\frac{1}{x}$. And also clearly is $(xy-1)\subseteq\ker(\phi)$. I just struggle to prove the other inclusion. Can ...
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18 views

What is the difference between ${\mathbb{Z}[x_1, .., x_n]}_{( p )}$ and ${\mathbb{Z}[x_1, .., x_n]}/(p)$?

Let $p$ be any prime. What is the difference between ${\mathbb{Z}[x_1, .., x_n]}_{( p )}$ and ${\mathbb{Z}[x_1, .., x_n]}/(p)$? ${\mathbb{Z}[x_1, .., x_n]}_{( p )}$ is the localization of ${\mathbb{Z}...
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On those integers $n>1$ such that any commutative ring with identity having exactly $n$ ideals is a PIR

Convention : All rings are commutative with unity unless stated otherwise. By ideals we will mean to include $\{0\}$ and $R$ also. Let us call an integer $n>1$ a "principal number" if any ring $R$...
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On those integers $n>1$ such that there exists a commutative ring with identity with exactly $n$ ideals

Let $n>1$ be an integer; we call $n$ a "ring number" if there exists a commutative ring $R$, with identity, having exactly $n$ ideals (including $\{0\}$ and $R$); now since for every $n>1$, $\...
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The second isomorphism theorem for C*-Algebras

in my functional analysis class right now we are studying the basics of C* Algebras and I was recently asked this question about the second isomorphism theorem for C* Algebras, but first let me cite ...
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$R$ be a commutative ring with unity such that every surjective ring homomorphism $f:R\to R$ is injective , then is $R$ Noetherian? [closed]

Let $R$ be a commutative ring with unity such that every surjective ring homomorphism $f:R\to R$ is injective , then is $R$ a Noetherian ring ?
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Primary decomposition in a Dedekind Domain

I was a little bit puzzled with the following problem that I have recently come across: Let $R$ be a Dedekind domain and let $P$ be a prime ideal in $R$. Is it true that $P^k$ is an irreducible ...
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71 views

Corollary to Lemma of Nakayama

In Matsumura's Commutative Algebra there is the following Corollary to the Lemma of Nakayama: Let $A$ be a ring, $M$ an $A$-module, $N$ and $N'$ submodules of $M$, and $I$ an ideal of $A$. Suppose ...
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Equivalences for a Banach lattice

I'm trying to prove the following equivalences for a Banach lattice $E$: $E$ has an order continuous norm Every monotone order bounded sequence in $E$ is convergent E is an ideal in $E^{**}$...
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33 views

$R$ be a finite commutative ring such that distinct ideals of $R$ have distinct orders (size) ; is $R$ a PIR?

Let $R$ be a finite commutative ring such that distinct ideals of $R$ have distinct orders (size) ; then is $R$ a Principal ideal ring (PIR) ? What if we moreover assume that distinct subrings of $R$ ...
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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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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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26 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 commutative ...
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46 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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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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21 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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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 $A_q= ...
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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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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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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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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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39 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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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 $ar\in I_i$...
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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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49 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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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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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 $$\sqrt{({X_{i_1}}^{a_1},...,{X_{i_k}}^{a_k})}=\sqrt{({X_{i_1}}^{a_1})+\cdots+({X_{i_k}}^{a_k})}=\...
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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 $$\sqrt{I}=\...
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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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41 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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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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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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41 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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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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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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40 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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23 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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76 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 k[...
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2answers
32 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, ...