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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1answer
25 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
24 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 ...
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0answers
133 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 ...
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0answers
81 views

Is $\operatorname{depth}(I,M)=\operatorname{depth}(S^{-1} I,S^{-1} M)?$ [closed]

Let $A$ be a ring, $I$ an ideal of $A$, and $M$ a finitely generated $A$-module such that $M\neq IM$. Show that there is a maximal ideal $\mathfrak m$ of $A$ such that ...
4
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0answers
69 views

Counterexamples for lcm-gcd identity and modular law for rings

In Mile Reid's Undergraduate Commutative Algebra, Exercise 1.3, we need to find counterexamples of lcm-gcd identity and modular law in the ring $A=k[X,Y]/(XY)$: $(I+J)(I\cap J)=IJ$; ...
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1answer
100 views
+50

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 ...
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1answer
57 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
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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
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?
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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 ...
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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. ...
3
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1answer
58 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 $a$. ...
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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 $\Bbb R[x,y]/J \cong \Bbb C$.
2
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0answers
107 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 ...
1
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0answers
40 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
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 ...
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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)= ...
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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) ...
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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 ...
1
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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 ...
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1answer
33 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 ...
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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 ...
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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
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1answer
44 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 ...
3
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0answers
24 views

Need help with finding generator

$I=\{a+bi \in R\mid a \equiv b\pmod 2\}$ is ideal of $R=Z[i]=\{a+bi\mid a,b \in Z\}$. Can somebody help me to find the generator of $I$?
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1answer
58 views

If $R/I$ satisfies Serre's condition $S_2$ then $R/(I,x)$ satisfies $S_1$

Let $I$ be an ideal of polynomial ring $R=K[x_1,\ldots,x_n]$ and $x$ be a non-zero divisor of $R/I$. Is the following statement true? If $R/I$ satisfies Serre's condition $S_2$ then $R/(I,x)$ ...
2
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1answer
38 views

A condition for a homogeneous ideal to be prime

The following is the problem 11 of Chaper 8 Section 4 of Ideals, Varieties, and Algorithms by Cox, Little and O'Shea. A homogeneous ideal is said to be prime if it is prime as an ideal in ...
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0answers
21 views

On maximal ideal spaces of a banach algebra

I am reading this article on maximal ideal spaces and there is this part that I don't quite understand very well, hope you guys can help me out. "Let $M(A)$ denote the maximal ideal space of a ...
1
vote
1answer
56 views

Let I,J ideals. Are they equal?

Let $$I= \langle 11x^5y+7xy^6+9,8xy^4+6xy+9 \rangle$$ $$J= \langle 7x^5y^2+17x^2y^5+29,13xy^4+62xy^3+19 \rangle$$ ideals. Examine whether those two ideals are equal. By seeing their 3D plots I ...
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votes
4answers
73 views

Maximal ideal in the ring of polynomials over $\mathbb Z$

Let $\mathbb Z[x]$ the ring of polynomials with integers coefficients in one variable and $I =\langle 5,x^2 + 2\rangle$, how can I prove that $I$ is maximal ideal. I tried first see that $5$ and ...
1
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1answer
42 views

Ideals problem from Hungerford Algebra

This question is from Hungerford's Algebra. Let $R$ be a ring with identity and $S$ the ring of all $n\times n$ matrices over $R$. $J$ is an ideal of $S$ iff $J$ is the ring of all $n\times n$ ...
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3answers
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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.
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1answer
34 views

A local subring of $F[[x]]$?

Suppose that $F$ is a field and $R=F⊕x^2F[[x]]$, where $F[[x]]$ is the ring of power series in one indeterminate $x$ with coefficients in $F$. I guess that $R$ is a local ring with the maximal ...
0
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0answers
28 views

Relation between the initial ideal and radical

Let $I$ be an ideal of the polynomial ring $S$. Show that ${\rm In}(\sqrt I)\subseteq\sqrt{{\rm In}(I)}$, where by ${\rm In}(I)$ we denote the ideal of initial forms of I, In(I) = (In(f) : f $\in$ I). ...
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0answers
67 views

Infinitely many maximal ideals containing $x^2-y^3$

How can we find infinitely many maximal ideals containing the ideal $I = (x^2-y^3)$ of $\mathbb C[x,y]$?
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1answer
26 views

Prove that if $\mathrm D$ is integral domain, then $\mathrm D$is UFD iff these conditions hold.

Prove that id $\mathrm D$ is integral domain, the "$\mathrm D$ is UFD " iff (1) $\mathrm D$ satisfies the ACC for principal ideals. (2) every irreducible element is a prime element. ...
3
votes
4answers
392 views

This ideal is prime

I'm trying to prove this ideal $$I=(x^2+y^2+x,x+y+xy)\subset \mathbb C[x,y]$$ is prime. I supposed that $I$ is prime and I'm using the classical method to prove $I$ is prime: If $ab\in I$, ...
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0answers
22 views

Complexity of and an algorithm for finding ideals of a ring?

One of the problems that has been a roadblock in my understanding of ideals has been how one would find them. One way of finding an I of some ring R would be to say $ \forall x \in I, \forall r \in R ...
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votes
1answer
34 views

ideal calculations: $2\mathcal{O}_K=\mathfrak{B}^4$ in the ring of integers of $K=\mathbb{Q}(i,\sqrt{2m})$

Let $K=\mathbb{Q}(i,\sqrt{2m})$ where $m \in \mathbb{Z}$ is odd and squarefree. Let $\alpha = (1+i)\sqrt{2m}/2$. Then $\alpha^2=im$, such that $\alpha$ is part of the ring of integers $\mathcal{O}_K$. ...
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1answer
27 views

Lie group and generated ideals

I have this question in my textbook, and I can't seem to solve it on my own: Let $P \subset GL(n,\mathbb{C})$ be a subgroup as following: $P$ consists of all matrices in block ...
2
votes
3answers
202 views

Is this example right (ideals of $\mathbb{Z}[x]$)?

I encountered the following problem: Let $I_{0}=\{f(x)\in \mathbb{Z}[x] \ | \ f(0)=0\}$. For any positive integer, show that there exists a sequence of ideals such that $I_0\subsetneq ...
3
votes
1answer
25 views

Ideal in Dedekind domain

Let $D$ be Dedekind domain and $I$ nonempty ideal in $D$. I have to show that there are only finitely many ideals $J$ in $D$ such that $I$ is contained in $J$. My first idea would be: assume that ...
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0answers
62 views

An $n$-generated ideal of grade $n$ can be generated by an $R$-sequence in any order

It is known to me that if $I$ is an $n$-generated ideal of a commutative ring $R$ with $\operatorname{grade}(I)=n$, then it is generated by an (ordered) $R$-sequence in $I$ of length $n$. I have a ...
0
votes
2answers
61 views

What are the elements like in $\mathbb C[x]/(2x^2+5)$?

As stated, I wonder what are the elements in $\mathbb{C}[x]/(2x^2+5)$ composed like? Since $2x^2+5$ is not monic, it seems to be a little different from the situation like in $\mathbb C[x]/(x^2+5/2)$. ...
4
votes
1answer
49 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: $$IJ=I\cap J$$ ...
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1answer
113 views

An ideal which is not maximal in $\mathbb{C}[x,y,z]$

Show that $$J=(x^2+y^2+z^2+x+y+z, x^5+y^5+z^5+2(x+y+z), x^7+y^7+z^7+3(x+y+z))$$ is not the maximal ideal $m=(x,y,z)$ in $\mathbb{C}[x,y,z]$.
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0answers
40 views

Universal Basis

How can I prove that $G=\{X-Y^2,XY-X,X^2-X,Y^2-Y^3\}$ is an universal Groebner Basis for the ideal $I=\{X-Y^2,XY-X\}$ in $\mathbb{Q}[X, Y]$ ? any suggestion is good. Thanks.
1
vote
1answer
16 views

Product of ideals is contained in intersection. Seemingly contradiction.

I am reading through a proof where the writer looks at a simple ring $A$ with a nonzero right ideal $M$. At a certain point he comes to this: The poduct ideal $AM$ is a two sided ideal, and so ...