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, ...

learn more… | top users | synonyms

0
votes
1answer
25 views

every z°-ideal I in C(X) is a rez°-ideal

I need to show that every z°-ideal I in C(X) is a rez°-ideal? in order to get a rez°-ideal I need to find an ideal an ideal J in C(X) such that I⊊J and I is a 〖z°〗_J-ideal. shall anybody help me in ...
0
votes
0answers
38 views

Show if $R$ is Noetherian, then $R_S$ is Noetherian [duplicate]

Show if $R$ is Noetherian, then $R_S$ is Noetherian. here is what I have read from somewhere else. Suppose $R$ is Noetherian and $J$ is an ideal $R_S$. Then $J=IR_S$ for some ideal $I$ of $R$. Since ...
0
votes
1answer
18 views

Show if $P$ is minimal prime ideal of $R$ then $PR_P$ is the only prime ideal of $R_P$

SHow if $P$ is minimal prime ideal of $R$ then $PR_P$ is the only prime ideal of $R_P$. Here are what I know and don't need to prove: I know $PR_P$ is the maximal ideal of $R_P$. I know $R_P$ is a ...
0
votes
2answers
86 views

Finding Radical of an Ideal [duplicate]

Given the ideal $J^\prime=\langle xy,xz-yz\rangle$, find it's radical. I know that the ideal $\langle xy,yz,zx\rangle$ is radical ideal but that's not the case. How can I compute the radical ...
1
vote
0answers
23 views

show if $P$ is minimal prime ideal of $R$ then every element of $PR_P$ is nilpotent.

Show if $P$ is minimal prime ideal of $R$ then every element of $PR_P$ is nilpotent. The only idea that I come to mind is, we know $PR_P$ is the maximal ideal of $R_P$. Since $P$ is a prime ideal of ...
1
vote
1answer
39 views

Coprime ideal definition

I am learning about ideals in my algebra class. If I have a ring $R$, I know that two ideals $I$ and $J$ in $R$ are coprime if $I+J=R$. I also know that $\mathbb{Z}$ is a principal ideal domain. I was ...
0
votes
2answers
41 views

If $Q$ is a radical ideal of $R$ with $Q∩S=∅$, then $QR_S$ is a radical ideal of $R_S$

If $Q$ is a radical ideal of $R$ with $Q∩S=∅$, show $QR_S$ is a radical ideal of $R_S$. Here is what I have done, since $Q$ is radical, let $q\in Q$ , then $q^n\in I$ for some $I$ ideal. let $a\in ...
0
votes
1answer
22 views

Is the localization of R by S is a subset of the ring R

Let $S$ be a multiplicatively closed subset of a commutative ring $R$. Then is it true that the localization $R_S=\{r/s:r\in R, s\in S\}$ a sub-ring of $R$? I think it is true, because $r/s=rs^{-1}$ ...
0
votes
1answer
98 views

Showing an ideal of $k[x,y]/\langle xy \rangle$ is prime

I am currently trying to show that the ideal $I = \langle x, y-1 \rangle$ is a prime ideal in $R = k[x,y]/\langle xy \rangle$ (for some field $k$). My first thought was to rewrite the ideal as ...
1
vote
1answer
76 views

If $\varphi(I)$ is an Ideal $\forall I $ ideal of $A$, is $\varphi$ surjective?

Today I heared some young students talking about the fact that if an homomorphism of rings (commutative with identity) $\varphi:A \rightarrow B$ is surjective then the image of any ideal of $A$ is an ...
3
votes
1answer
56 views

Find a non-principal ideal in $ \Bbb Z [2i]$.

Find a non-principal ideal in $ \Bbb Z [2i]$. I think it might be $(1+2i,1-2i)$, but have problems with proving this. I know that $|1+2i|=|1-2i|=5$. Moreover, there are only 6 elements with ...
0
votes
3answers
146 views

Find prime ideals of the ring $\Bbb Z [ \sqrt[3]2]$ which contain $5$

Find all prime ideals $p$ of the ring $ R= \Bbb Z [ \sqrt[3]2] $ such that $ 5 \in p$ and find $R/p$ for all of them. I know that of course $R$ is prime and $R/R = \{0 \}$. Unfortunately I have no ...
0
votes
1answer
35 views

What does the square-zero ideal means

I tried to study why Krull's intersection theorem won't work in non-Noetherian rings. It was said here that the example written by user2035 works by taking some kind of square-zero ideal. How do one ...
3
votes
2answers
119 views

Maximal Ideal Spaces

STATEMENT: Let $C_b(\mathbb{R})$ be the $C^*$-algebra of bounded continuous functions on $\mathbb{R}$. Let $A$ be the $C^*$-subalgebra of $C_b(\mathbb{R})$ generated by $C_∞(\mathbb{R})$ together with ...
2
votes
1answer
28 views

$I=(I:s)\cap (I, s)$

Somewhere I've read the following: Theorem Let $I \subset A$ an ideal of a domain $A$. Let $S$ a multiplicatively closed set and let be $I^e$ the image of $I$ in $S^{-1}A$. Let $s \in S$ be such ...
3
votes
3answers
39 views

Verifying that a given set is an ideal

I am trying to show that I have said: f=0 is in I so I is non-empty. Let $f,g$ be members of $I$ $(f+g)(\sqrt5)=f(\sqrt5)+g(\sqrt5)=0+0=0 ==> f+g ∈ I$ so I is closed under addition Let ...
0
votes
1answer
22 views

Ideals and quotient rings [duplicate]

I am trying to show that (R/I)/(J/I) is isomorphic to R/J I and J are both ideals of the ring R, and I is a subset of J. How do I begin this proof?
1
vote
1answer
63 views

Does taking quotients preserve isomorphism of rings?

Let $R$ be a commutative ring and $A$ and $B$ be subrings of $R$. Suppose also that an ideal $I$ of $R$ is contained in both $A$ and $B$ (so $I$ is an ideal of both $A$ and $B$). I have two ...
2
votes
1answer
22 views

trying to show the kernel of a map is an ideal

Let $F : \mathbb{R}[x] \to \mathbb{C}$ be given by $F( f(x) ) = f(i) $ Want to show $\ker F = \langle x^2 + 1 \rangle$ (ideal generated by $x^2+1$) . MY attempt: If $f \in \langle x^2 + 1 \rangle ...
0
votes
0answers
39 views

$IM=mM$. can we say that $I$ is a reduction ideal of $m$

Definition. Let $R$ be a Noetherian ring􀀀, $I$ a proper ideal,􀀀 and $M$ a finite $R$-module. An ideal $J\subset I$ is called a reduction ideal of $I$ with respect to $M$ if $JI^nM = I^{n+1}􀀀M$ for ...
3
votes
1answer
70 views

Show ideals are equal

Blockquote Let $k$ be a field. Let $I \subset k[x_1,...,x_n]$ be an ideal and let $f_1,...,f_s \in k[x_1,...,x_n]$. Using the fact that the following are equivalent i) $f_1,...,f_s \in I$ ii) ...
0
votes
1answer
49 views

Let K be a field, and $I=(XY,(X-Y)Z)⊆K[X,Y,Z]$. Prove that $√I=(XY,XZ,YZ)$.

Let $K$ be a field, and let $I=(XY,(X-Y)Z) \subset K[X,Y,Z]$. Prove that $\sqrt{I}=(XY,XZ,YZ)$. I have no idea how to start with this question, can anybody give me some hint? Thanks a lot.
0
votes
1answer
37 views

Show that $Z(XY,XZ,YZ)$ is not irreducible

Show that $Z(XY,XZ,YZ)$ is not irreducible. what I think it is $Z(XY,XZ,YZ)=Z(XY)∩Z(XZ)∩Z(YZ)=(Z(X)∪Z(Y))∩(Z(X)∪Z(Z))∩(Z(Y)∪Z(Z))$ Then I am not sure how to carry on, and what I need to show it is ...
0
votes
1answer
19 views

Prove that for ideals I and J of a commutative ring, $√(I∩J)=√(IJ)$

Prove that for ideals $I$ and $J$ of a commutative ring, $√(I∩J)=√(IJ)$, where $√(I∩J)$ and $√(IJ)$ are the radical ideals. The first step I use the idea that $IJ\subseteq{I∩J}$ is true for all ...
0
votes
1answer
47 views

Question on prime ideals of ${\mathbb Z}[x]$

I was thinking this question myself: Consider the topological space $(\text{Spec}(\mathbb{Z}[x]),T )$ where open sets $D_I$ in $T$ are given as indexed by ideals $I$ in $\mathbb{Z}[x]):$ $D_I =\{p\in ...
1
vote
1answer
33 views

Frobenius powers of an ideal does not depend on the choice of a system of generators

Let $I$ = $(x_1 , . . . , x_n )$ be an ideal of a ring $R$ of characteristic $p$. For each nonnegative integer $e$ we set $I^{[p^e]}$=$(x_1^{p^e},...x_n^{p^e}$)$R$. These ideals are called the ...
0
votes
0answers
14 views

Inclusion with some principal ideals

Let $A$ be a principal ideal domain, $M$ a free $A$-module of rank $n$, $M'$ a submodule of $M$ with $M' \ne (0)$, and $L(M,A)$ the set of linear forms on $M$. For $v \in L(M,A)$, we can write ...
2
votes
2answers
42 views

Describe $\mathbb{Z}[\omega] / (2)$

$$\Bbb Z[ω] = \{\;a + bω: a, b\in\Bbb Z\;\}\;,\;\; ω = e^{2πi / 3} = -\frac12 +\frac{\sqrt3}2i$$ Describe $\;\Bbb Z[ω]/(2)\;$ where $\;(2)\;$ is an ideal. I already described it as a ring, and am now ...
1
vote
0answers
58 views

Chain of ideals in nilpotent algebra

Let $R$ be a nilpotent algebra ($R^n = \{0\}$ for some $n \ge 1$) and $A$ be a subalgebra of $R$. I want to show that exist a finite chain of subalgebras {$R_i$ | $i = 0, 1, ..., m $}, $m \ge 1$, ...
1
vote
1answer
38 views

If there exists a vertex of $ \Gamma_{2}(R)\setminus J(R) $ which is adjacent to every other vertex then $ R \cong \mathbb{Z}_{2}\times F$

I am reading the research paper Comaximal Graph of Commutative Rings by H.R. Maimani, M. Salimki, A. Sattari, S. Yassemi. In this paper, $ R $ denotes a commutative ring with the identity element. $ ...
0
votes
0answers
20 views

Elementary Divisor Ring

Let R Hermite ring. I think if I can prove if "R/I elementary divisor ring then R elementary divisor ring", I can prove a theorem in my thesis. Anyone can help me? Please.
1
vote
1answer
59 views

if R is a commutative ring in which all the prime ideals are finitely generated then R is Noetherian [duplicate]

Prove that if $R$ is a commutative ring in which all the prime ideals are finitely generated, then $R$ is Noetherian. Here is what I been told to do: Suppose that $R$ is not Noetherian, and use ...
1
vote
2answers
41 views

Preimage of prime ideal under induced map

Let $\phi: R \to S$ be a ring homomorphism. Let $\phi^*:Spec(S) \to Spec(R)$ is induced map of sets. Here $Spec(S)$ is the set of prime ideals of a ring. Is $\phi^*$ surjective? I think that it's ...
1
vote
2answers
69 views

The finitely generated-ness of ideals $I +rR$ and $I:r$ imply $I$ is a finitely generated ideal [closed]

Let $I$ be an ideal of a commutative ring $R$, and let $r ∈ R$. Show that if the ideals $I +rR$ and $I:r=\{s∈R:sr∈I\}$ are finitely generated, then $I$ is a finitely generated ideal. Can anyone give ...
0
votes
1answer
51 views

Proof check: find all Prime ideals of $R[T]/\langle x^n\rangle$,

Given $R=\mathbb{C}[x]/\langle x^n\rangle$, each element in $R$ may be represented as $a_0+a_1x+\cdots+a_{n-1}x^{n-1}$. I'm guessing that $P=\langle x^i\rangle$ for $i=\{1,\ldots,(n-1)\}$ represents ...
1
vote
1answer
41 views

Given a radical ideal, prove that the ideal cannot be prime

Let $I\subseteq K[x_1,...,x_n]$ be an ideal. Let $\sqrt{I}$ be the radical of the ideal. Assuming that $I$ is radical, i.e $I=\sqrt{I}$, show that $I$ is not prime only when there exist ideals $G,H$ ...
0
votes
0answers
15 views

Functions which 'respect' Ideals

Let $R$ be a ring such that $|R| = |\mathbb{N}|$. Let $I_R$ be the set of ideals in $R$. We define $$ S_J = \{f:R\ \rightarrow \ R\ | \ \forall a,b \in R: \ a + J = b+ J \ \Rightarrow \ f(a) + J = ...
2
votes
1answer
28 views

$V$ is irreducible exactly then when $I(V )$ is a prime ideal

If $V$ is an algebraic set of $K^n$, show that $V$ is irreducible exactly then when $I(V )$ is a prime ideal of $K[X_1, X_2, \dots, X_n]$ . Let $V$ be irreducible. We suppose that $I(V)$ is not a ...
2
votes
1answer
116 views

Can we use the Nullstellensatz?

In $\mathbb{C}[x, y, z]$ we have that $V=\{y-x^2, z-x^3)=\{(t, t^2, t^3) | t \in \mathbb{C}\}$. To show that $$I(V(y-x^2, z-x^3))=\langle y-x^2, z-x^3\rangle $$ can we use the Nullstellensatz?? ...
0
votes
1answer
37 views

Why is that the radical ideal?

In my lecture notes we have the following: Definition: $f, g \in \mathbb{C}[x, y]$ $f \sim g \Leftrightarrow \exists c \in \mathbb{C}, c \neq 0$ such that $g=cf$ Example: If $f \sim g ...
2
votes
1answer
29 views

$V_1=V(x-y)$ and $V_2=V(x+y)$ are algebraic sets

I am looking at irreducible algebraic sets. $V \subseteq K^n$ is an algebraic set $\Leftrightarrow$ it is of the form $V(I)$, where $I=$radical Ideal of $K[x_1, x_2, \dots , x_n]$. At my lecture ...
0
votes
1answer
18 views

Algebraic set - Radical Ideal - $Rad(Rad(I))=Rad(I)$

In my lecture notes we have the following: $V \subseteq K^n$ is an algebraic set $\Leftrightarrow$ it is of the form $V(I)$, where $I=$radical Ideal of $K[x_1, x_2, \dots , x_n]$. It stands that ...
2
votes
1answer
35 views

Algebraic Set-Radical Ideal-Nullstellensatz

In my lecture notes there is the following: $$I \rightarrow V(I) \rightarrow I(V(I))$$ It stands that in general $I \subsetneq I(V(I))$. The equality stands if and only if $I$ is a radical ...
1
vote
1answer
26 views

Proof of the proposition $V(S)=V(\langle S \rangle )$

In my lecture notes we have the following: Proposition: $$V(S)=V(\langle S \rangle )$$ Proof: $$\langle S \rangle=\left \{\sum_{i=1}^m g_i f_i | f_i \in S, g_i \in R=K[x_1, x_2, \dots , ...
0
votes
1answer
31 views

Operations with ideals: sum and product

Operations at ideals. The sum is defined as $$I_1 + I_2 + \dots + I_m =\{a_1+a_2+\cdots +a_m\mid a_i \in I_i\}.$$ It can be proven that $$I_1 + I_2 + \dots + I_m \trianglelefteq R$$ and each ...
1
vote
2answers
56 views

$Rad(I)$ is an ideal of $I$

$$Rad(I)=\{a \in R | \exists n \in \mathbb{N} \text{ such that } a^n \in I\}$$ R is a commutative ring, I is an ideal. To show that $Rad(I)$ is an ideal of $I$, we have to show that for $a,b \in ...
0
votes
1answer
25 views

For a maximal ideal $M$ of $R$ of a commutative ring $R$ ( not necessarily with unity ) , then is $R/M$ a simple ring ?

Let $M$ be a maximal ideal of a commutative ring $R$ ( not necessarily with unity ) ; then is it true that the only ideals of $R/M$ are the trivial ones i.e. is it true that $R/M$ is a simple ring ? ...
0
votes
0answers
25 views

No such prime ideal contains $I_1+I_2\implies I_1 $ and $I_2$ are relatively prime

It's clear to me that if $I_1$ and $I_2$ are two relatively prime ideals of a ring $R$, then there is no such prime ideal containing $I_1+I_2$, since by definition of relatively prime ideals ...
0
votes
2answers
40 views

$I$ is maximal ideal $\implies$ $R/I$ has no proper ideals

I'm reading through a proof in a book on commutative algebra and in the proof it uses the fact that $I$ is a maximal ideal $\implies$ $R/I$ has no proper ideals, by using the correspondence theorem. ...
0
votes
3answers
91 views

$\mathbb{Z}_{(2)}$ has one maximal ideal

My lecture notes state that the set $\mathbb{Z}_{(2)}$, defined as $$\mathbb{Z}_{(2)}:=\left\{\frac{a}{b}\in\mathbb{Q}\mathrel{}\middle|\mathrel{}\gcd(a,b)=1\text{ and } 2\nmid b\right\}$$ has a ...