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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2
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3answers
117 views

Number of elements in the ring $\mathbb Z [i]/\langle 2+2i\rangle$

The question is : Show that $I=\langle 2+2 i\rangle$ is not a prime ideal of $\mathbb Z[i]$. Also find the number of elements in $\mathbb Z[i]/I$ and its characteristic. My try: I started with ...
31
votes
6answers
3k views

Quotient ring of Gaussian integers

A very basic ring theory question, which I am not able to solve. How does one show that $\mathbb{Z}[i]/(3-i) \cong \mathbb{Z}/10\mathbb{Z}$. Extending the result: $\mathbb{Z}[i]/(a-ib) \cong ...
-4
votes
0answers
64 views

Transcendence degree of Rees ring [on hold]

Let $R$ be a ring which is a domain and $I$ an ideal of $R$. How can I compute the tr.deg of the Rees ring $R(I)$ over $R$? In this way I want to check the altitude formula.
2
votes
0answers
20 views

Finding ideal representatives in the class group of $\mathbb{Q}(\zeta_{23})$

I know that $\mathbb{Q}(\zeta_{23})$ has class number 3, and I am wondering how I can find ideal representatives of the two nonprincipal classes in the class group. I have tried looking at examples ...
13
votes
1answer
108 views
+50

Is there a geometric meaning of a prime power not being primary?

I guess that the standard example of a prime power that is not a primary ideal is $$\mathfrak p^2 :=(x,z)^2\subset k[x,y,z]/(xy-z^2):=A.$$ Because $\mathfrak p^2 = (x^2,xz,xy)$, we see that $x\not ...
3
votes
0answers
105 views

Counterexamples for lcm-gcd identity and modular law for rings

In Miles Reid's Undergraduate Commutative Algebra, Exercise 1.3, we need to find counterexamples of lcm-gcd identity and modular law in the ring $A=k[X,Y]/(XY)$: $(I+J)(I\cap J)=IJ$; ...
1
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0answers
59 views

Ideals of $\Bbb Z/p^2q\Bbb Z$

Let $p,q$ be distinct primes. Then $\mathbb{Z}/p^2q\mathbb{Z}$ has 3 distinct ideals. $\mathbb{Z}/p^2q\mathbb{Z}$ has 3 distinct prime ideals. $\mathbb{Z}/p^2q\mathbb{Z}$ has 2 distinct prime ...
0
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0answers
34 views

Coheight of an ideal

I am considering a quotient ring $R=\mathbb F_2[x_1,\dots,x_n]/I$ that is Cohen-Macaulay but not local and an ideal $J$ in $R$. If $R$ were local, then one had the equality $$\mathrm{coheight}(J)=\dim ...
1
vote
1answer
15 views

Expressing a hereditary subalgebra in terms of a state

Let $\mathcal{A}$ be a C*-algebra and $\phi$ a state on $\mathcal{A}$. Then, it's not hard to see that $\mathcal{L} = \left\{ x : \phi(x^*x)=0 \right\}$ is a closed left ideal of $A$ and so ...
4
votes
1answer
143 views

Is there an example of commutative ring with exactly three prime ideals for which this property holds?

Is there an example of commutative ring with exactly three non zero prime ideals $P_i$ which satisfies the following statement: $P_1P_2=0$ and for an ideal $I\neq 0$ such that $I\neq P_i$ we have ...
1
vote
1answer
81 views

Prime ideals in formal power series

Let $A$ be a commutative ring with unit. If $\mathfrak{p} \subset A $ is a prime ideal, then $\mathfrak{p}$ is the contraction of a prime ideal of $A[[x]]$, the ring of formal power series. Why is ...
2
votes
1answer
57 views

The geometric interpretation for extension of ideals?

Suppose $f\colon B\to A$ is a ring homomorphism, and $I\subseteq B$ is an ideal. What's the geometric interpretation for the extension $f(I)A$ of the ideal $I$? Especially, I'm interested in the case ...
1
vote
2answers
76 views

Suppose that R is a commutative ring with unity such that for each $a$ in $R$ there is an integer $n > 1\mid a^n =a$. Every prime ideal is maximal?

Suppose that R is a commutative ring with unity such that for each $a$ in $R$ there is a positive integer $n > 1$ such that $a^n =a$. Prove that every prime ideal of $R$ is a maximal ideal of R. ...
3
votes
3answers
139 views

Maximal ideal in the ring of functions from $\mathbb{R} \to \mathbb{R}$

Well, the problem I'm trying to solve is this: Let A be the ring of all continuos functions from $\mathbb{R} \to \mathbb{R}$. Show that $$M = \{f \in A; f(0)=0\}$$ is a maximal ideal of A. I tried ...
2
votes
3answers
50 views

In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal. What could be wrong in this approach?

In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal Attempt: Let $R$ be the principal ideal domain. A principal ideal domain $R$ is an integral domain in which ...
0
votes
1answer
37 views

Suppose that $I$ is an ideal of $J$ and that $J$ is an ideal of $R$. prove that if $I$ has a unity then $I$ is an ideal of $R$

Suppose that $I$ is an ideal of $J$ and that $J$ is an ideal of $R$. prove that if $I$ has a unity then $I$ is an ideal of $R$ Attempt: Given that $I$ is an ideal of $J$ which means : ...
-6
votes
2answers
276 views

Quotient ring isomorphism

I think that if $A$ is any commutative ring with unity and $q\in A$, $p\in A[x]$ then we have $A[x]/(q,p)\cong A/(q)[x]/(\bar{p})$ where $\bar{p}$ denotes the class of $p$ in $A/(q)[x]$. Is this true? ...
2
votes
1answer
28 views

Prime radical that is nil but not nilpotent

Please help me how to show that the prime radical of the ring $R=\prod\limits_{n = 1}^\infty { \mathbb{Z} /2^n\mathbb{Z} } $is nil but not nilpotent.
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1answer
59 views

Given ring and ideal, How to prove that the intersection of ideals is an ideal

Given $R$ is a ring, $X\subseteq H_i$ and $H_i$ is an ideal of $G$ for each $i=1,2,...,n$. Prove that $H_1∩H_2∩...∩H_n$ is an ideal of $G$ and contain $X$. That is a question I get from random ...
4
votes
1answer
347 views

Is the number of prime ideals of a zero-dimensional ring stable under base change?

Let $A$ be a zero-dimensional ring of finite type over a field $k$ and let $X= \textrm{Spec} \ A$ be its spectrum. Note that $X$ is a finite set. Suppose that $k\subset K$ is a finite field extension ...
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0answers
140 views

If $A$ is complete for $I$-adic and $J$-adic topologies, then $A$ is also complete for the $(I+J)$-adic topology

If $A$ is complete for both $I$-adic and $J$-adic topologies, then $A$ is also complete for the $(I+J)$-adic topology. (Matsumura, CRT, Exercise 8.1) How can I solve this problem? A is a ring ...
3
votes
1answer
421 views

$I$-adic completion

Let $A$ be a commutative noetherian ring, and suppose that $A$ is $I$-adically complete with respect to some ideal $I\subseteq A$. Is it true that for any ideal $J\subseteq I$, the ring $A$ is also ...
2
votes
1answer
155 views

If a local ring is $\mathfrak{m}$-adically complete is it also $I$-adically complete [duplicate]

Suppose $(R,\mathfrak{m})$ is a local ring and $I$ a proper ideal. If $R$ is $\mathfrak{m}$-adically complete is it also $I$-adically complete.
0
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1answer
25 views

The nil-radical is an intersection of all prime ideals proof

Every proof I found online made the same implications. Take one for example: http://www.artofproblemsolving.com/Wiki/index.php/Nilradical I'm quoting the relevant part, which confuses me: "To show ...
4
votes
1answer
35 views

Quotient $M/M^2$ is finite dimensional over $R/M$ in local Noetherian ring?

I have that $R$ is a Noetherian local ring with maximal ideal $M$, and I want to show that $M/M^2$ is a finite dimensional vector space over the field $R/M$. I think I've proved this (though I ...
4
votes
0answers
149 views

Primary decomposition of ideals

How to find a primary decomposition of the ideal $I = (X^2, XY, XZ, YZ)$ in the ring $k[X,Y,Z]$? Is there a general rule for finding primary decompositions? Also how to show that $(X,Y)^{308}$ is a ...
5
votes
4answers
292 views

Idempotents in a local ring

Is it true that a local ring, i.e., a commutative ring with a unique maximal ideal, doesn't contain idempotent elements $\neq 0, 1$ ? Why ? Any hint ?
3
votes
1answer
59 views

Locally unital ideals [duplicate]

Let $R$ be a ring with unity not necessarily commutative and $I$ an ideal of $R$. Let for every element $a \in I$ there exists an element $c\in I$ such that $ac=a$. Note that $c$ is related to $a$. ...
4
votes
1answer
50 views

What kind of rings have exactly three ideals?

What kind of rings(commutative, w/ unity) have exactly three ideals? I know that those with exactly two ideals are "the fields", but what about three? Is there a fancy name for them?
0
votes
1answer
17 views

Finding a particular principal open subset of $Spec R$

Let $V\subseteq U$ be open subsets of $X=\text{Spec } R$, where $R$ is a commutative ring. So $V$ is the set of prime ideals not containing some ideal $I$, and $U$ is the set of prime ideals not ...
5
votes
3answers
121 views

Commutative ring with an ideal that contains all the nonunits

Is there an example of a commutative ring with an ideal that contains all the non-units? I was trying to think of some subring of $\mathbb Q$, but I couldn't get it to work.
4
votes
1answer
50 views

Do $IJ$ and $I\cap J$ coincide if $I$ and $J$ are coprime? Also if ring $R$ has a $1$ and is not commutative?

Let $R$ be a ring (with identity) and let $I,J$ be two coprime (two-sided) ideals in it. In Algebra: Chapter $0$, Aluffi, III. exercise 4.5. the reader is asked to prove that: $$IJ=I\cap J$$ ...
-2
votes
0answers
23 views

locally unital ideals of ringa [duplicate]

1 down vote favorite Let R be a ring with unity not necessarily commutative and I an ideal of R.Let for every element a of I there exists an element c of I such that ac=a.Note that c is related to a. ...
2
votes
0answers
111 views

Reduced Gröbner basis and extension of scalars

Consider a field extension $L\subseteq K$, and let $\mathfrak a\neq 0$ be an ideal of the polynomial ring $L[T_1,\ldots,T_n]$. Suppose that a monomial order is fixed, so there exists a unique reduced ...
0
votes
0answers
38 views

Prime ideal is contraction of prime ideal iff it's saturated

Let $\varphi: A\to B$ be a commutative ring homomorphism and $P$ a prime ideal of $A$. The expansion of an ideal $I\subset A$ is the ideal generated by $\varphi(I)$ in $B$, and the contraction of an ...
2
votes
1answer
34 views

Which is a subring? Which is an ideal?

We are having ring $\mathbb{Z}[\sqrt{-6}]$. Which of the sets is subrings of $\mathbb{Z}[\sqrt{-6}]$ and which are ideals? $\mathbb{Z}+5\mathbb{Z}[\sqrt{-6}]$ $5\mathbb{Z}+\mathbb{Z}[\sqrt{-6}]$ ...
0
votes
2answers
50 views

Maximal ideal which isn't principal

Let $J=(x-2,x-y^2-3)$ ideal in the polynomial ring $\Bbb R[x,y]$. Please help me prove that $J$ is a maximal ideal which isn't principal, and that $\Bbb R[x,y]/J \cong \Bbb C$.
0
votes
1answer
24 views

Specific Ideal determinations for a Ring

Uploaded in a picture, rather than typing it all. Note that these are unmarked questions from a sample exam. Just trying to study, but have forgotten almost everything (three major exams before this ...
0
votes
2answers
64 views

A question about the Zariski Topology

Let $\{a_i\}$ be an infinite set of ideals in commutative ring $R$. Is $\bigcap\limits_{i=1}^\infty a_i$ not defined? I am trying to understand Zariski Topology. Here, $V(\bigcap a_i)= ...
0
votes
1answer
33 views

A question about ideals.

Let $A$ and $B$ be arbitrary subsets of a ring. Then $V(A\cup B)=V(A)\cap V(B)$. Here, $V(X)$ is the set of prime ideals containing $X$. Let $W(X)$ be the set of ideals (any sort of ideals) ...
0
votes
0answers
61 views

What are the open and closed sets in $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)$?

What are the open and closed sets in $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)$ ? $\operatorname{Spec}\mathbb{C}[x,y]/(y^2-x^3)=\{ (0),\ (\tilde{x}-a,\tilde{y}-b),\ b^2=a^3\}$.
1
vote
1answer
16 views

Determining if any of these three are an ideal of $\mathbb{R}[x]$

$\mathbb{R}[x]$ denotes the ring of polynomials in $x$ with real coefficients. Let $I \subset \mathbb{R}[x]$ be the subset of those polynomials with constant coefficient $0$, and let $J \subset ...
0
votes
0answers
37 views

Grade of an ideal greater than the projective dimension of quotient of another one

We know that the grade of an ideal $I$ in a Noetherian ring $R$ is the infimum of the set of all $i$ with $Ext^i(R/I,R)$ nonzero. Also, the projective dimension of an $R$-module $M$ is at most $s$ if ...
0
votes
1answer
37 views

Help to conceive a name

Filter $F$ is defined by the formula $$A\cap B\in F \Leftrightarrow A\in F\wedge B\in F.$$ Ideal $F$ is defined by the formula $$A\cup B\in F \Leftrightarrow A\in F\wedge B\in F.$$ In my book I ...
1
vote
1answer
34 views

Proving $M_p$ is maximal in $C[0,1]$

Let $M_p$ be the ideal of those continuous functions of $C[0,1]$ which have $p\in [0,1]$ as a zero. It is a commonly known fact that $M_p$ is a maximal ideal. However, the proof is generally ...
1
vote
1answer
109 views

Subset of $\mathbb{Z} \times \mathbb{Z}$

I have a past exam question that is as follows: Let $k$ be a fixed integer and $S = \{(a,ka)|a \in \mathbb{Z}\}$ be a subset of $\mathbb{Z} \times \mathbb{Z}$. Prove that $S$ is a subgroup of ...
1
vote
1answer
146 views

Primary ideals in Noetherian rings

For an $R$-module $M$ I have the following definition for a submodule $N\subset M$ to be $\mathfrak{p}$-primary: this is the case when $\text{Ass}(M/N) = \{\mathfrak{p}\}$, that is, $M/N$ is coprimary ...
1
vote
1answer
45 views

Every proper ideal $I$ in a nonzero commutative unitary ring $R$ is contained in a maximal ideal.

If $R$ is a nonzero commutative unitary ring, then $R$ has a maximal ideal. Indeed, every proper ideal $I$ in $R$ is contained in a maximal ideal. There is a proof of this in Rotman's Advanced ...
4
votes
1answer
208 views

Set of all compact operators $K(H)$ is the unique ideal in $B(H)$?

I want to show that the set of all compact operators $K(H)$ is the unique ideal in $B(H)$. Is there any relation between invertibility and compactness of an operator?
1
vote
1answer
42 views

Annihilator of a quotient module

Let $J$ be an ideal of $R$, and $M$ a right $R$-module. Since $Jr \subseteq J$, $M / MJ$ is naturally a right $R$-module. Since it seems relevant to another problem, I am trying to determine ...