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

1
vote
1answer
26 views

Prove that $f:I\oplus J\to R\oplus IJ: (i,j)\mapsto (i+j,ij_0-i_0j)$ is surjective for coprime $I,J$

I have to prove that if $I,J$ are coprime ideals in a commutative ring $R$, and $i_0\in I, j_0\in J$ are s.t. $i_0+j_0=1$, then $$f:I\oplus J\to R\oplus IJ: (i,j)\mapsto (i+j,ij_0-i_0j)$$ is ...
2
votes
1answer
62 views

Determining the structure of the quotient ring $\mathbb{Z}[x]/(x^2+3,p)$

I'm interested in the following problem from Artin's Algebra text: Determine the structure of the ring $\mathbb Z[x]/(x^2 + 3,p)$, where (a) p = 3, (b) p = 5. I know that by the isomorphism ...
8
votes
3answers
97 views

Any left ideal of $M_n(\mathbb{F})$ is principal

I'm working on the following problem: Let $A$ be the ring of $n \times n$ matrices over a field $\mathbb{F}$. (a) Show that for any subspace $V$ of $\mathbb{F}^n$, the set $I_V$ of matrices ...
1
vote
0answers
53 views

If $JA_{P}\subset IA_{ P}$ for every associated prime, then $J\subset I$

This problem is Exercise 6.4 in Matsumura's Commutative Ring Theory. Let $I$ and $J$ be ideals of a Noetherian ring $A$. Prove that if $JA_P\subset IA_{P}$ for every $P\in ...
2
votes
3answers
110 views

Is $\mathbb{Z}[x]\over \langle x+3\rangle$ field?

Is $\mathbb{Z}[x]\over \langle x+3\rangle $ field? Can I say $x+3$ is irreducible over $\mathbb{Z}$, so it is field.
3
votes
1answer
86 views

$R$ be a commutative ring with unity satisfying a.c.c. on radical ideals ; is it true that $R[x]$ also satisfies a.c.c. on radical ideals ?

$R$ be a commutative ring with unity satisfying ascending chain condition on radical ideals ; is it true that $R[x]$ also satisfies ascending chain condition on radical ideals ?
0
votes
1answer
31 views

Why is the following affine varieties equal? “Closure theorem”

I am very unsure about this "closure theorem" of ideals and varieties. I'm not sure if anyone here can answer this concisely as I think some notations may differ from what others know... Say $I$ is ...
4
votes
3answers
70 views

Ideal of $\mathbb Q[x]$ which contains two polynomials

Suppose $I$ is an ideal of $\mathbb Q[x]$ which contains $x^2 + 2x +4$ and $x^3 - 3$. Prove $I =\mathbb Q[x]$. This is an exercise in my abstract algebra text book. I know the definition of an ...
4
votes
2answers
86 views

Invertible ideals are finitely generated.

Let $R$ be an integral domain and let $I,J \subseteq R$ be ideals. Suppose $IJ=(a)$ for some $a \in R$. We wish to show that $I$ and $J$ are finitely generated. Since $a \in IJ$ we know $a$ can ...
1
vote
3answers
67 views

Prove that a proper ideal of R is a maximal ideal of R, with R being a commutative ring with unity

R is a commutative ring with unity and A is a proper ideal of R, and every element of R that is not in A is a unit of R. Prove that A is a maximal ideal of R. Can anyone give a head start on how to ...
0
votes
0answers
40 views

Modular reduction on ideals generated by two elements

I know that $\mathbb{Z}[x]$ is a commutative ring with unity and for any two elements $f(x)$ and $g(x)$, the set $\langle f(x), g(x)\rangle = \{ \alpha(x)f(x) + \beta(x)g(x) : \alpha(x) \land ...
1
vote
0answers
17 views

Product of two nilpotent subideals is nilpotent subideal

Supposse $X,Y$ are nilpotent subideals of algebra $A$, i.e. such nilpotent subalgebras of $A$ that: $$ \exists \{X_i\}: \quad \{e\}\lhd X=X_1 \lhd X_2\lhd\ldots\lhd X_n=A $$ $$ \exists \{Y_j\}: ...
3
votes
2answers
39 views

Prime ideals in $R= \left\{ \frac ab\in \mathbb Q\mid a,b\in \mathbb Z,p\nmid b \right\}$

Let $p\in \mathbb Z$ be a prime. Define $R= \left\{ \frac ab\in \mathbb Q\mid a,b\in \mathbb Z,p\nmid b \right\}$. I'm supposed to prove $pR$ is the only prime ideal in $R$, and the $R/pR\cong \mathbb ...
3
votes
1answer
68 views

$\langle 7, 3 + \sqrt{-5} \rangle = \langle 7, 4 + \sqrt{-5} \rangle$, right?

I'm not in a math class (haven't been in years) but if this question about $\textbf{Z}[\sqrt{-5}]$ appears in some textbook, I wouldn't be surprised. What I have done: $$1 \times 7 + (3 - ...
1
vote
1answer
19 views

A commutative ring with at most $5$ distinct ideals is a PIR

Let $R$ be a commutative ring with unity having at most $5$ distinct ideals (including $\{0\}$ and $R$ itself); then is it true that $R$ is a principal ideal ring i.e. is every ideal of $R$ principal? ...
1
vote
1answer
28 views

Relating generating sets of modules over quotient ring to modules over original ring

I am having trouble with the following exercise. Let $R = F[x]$, where F is some field. Let $I = (x^2)$, and let M be an R/I-module (induced action, so assuming $IM = 0$). Then I want to show that ...
1
vote
0answers
34 views

Radical ideals and prime decompositions for nonprincipal ideals

Fact. Let $R$ be a UFD. A principal ideal $ \left\langle a \right\rangle$ is radical iff every element in its prime decomposition has multiplicity one. Does this statement generalize in any way to ...
0
votes
1answer
76 views

$\mathrm{Ann}(I \cap J) = \mathrm{Ann}(I) +\mathrm{Ann}(J)$ if $R$ is self-injective

If $R$ is injective as an $R$-module, then for every two ideals $I$ and $J$ we will have $\mathrm{Ann}(I \cap J) = \mathrm{Ann}(I) + \mathrm{Ann}(J)$. I made an effort for an hour but it was not ...
1
vote
2answers
37 views

What is meant by saying $I=(5a+b\sqrt{5}) $ is a principal ideal

Hi I am wondering if someone can explain something in my notes. It says, consider $$\mathbb{Z}(\sqrt{5})=\{a+b\sqrt{5}:a,b \in \mathbb{Z}\}$$ then $$I=\{5a+b\sqrt{5}: a,b \in \mathbb{Z} \}$$ is an ...
1
vote
1answer
78 views

Please help with question about an ideal in $\mathbb{Z}$

Hi I am needing some help and guidance on determining how to know when something is an ideal. We are working with commutative rings. Our definition of an Ideal I, of a commutative ring R , is a ...
1
vote
1answer
45 views

Prove the equality of two ideals in $F[x]$

Prove $\langle x-1\rangle = I$ where $I= \{ a_0 x ^0 +\dots +a_n x^n: a_0+\dots +a_n=0\}$ in $F[x]$, $F$ a field. Proof we need to show $\langle x-1\rangle \subseteq I$ and $I \subseteq \langle ...
1
vote
1answer
26 views

Maximal ideal in commutative Banach algebras. Why commutative?

I am having some trouble in understanding where is used the fact that the algebra is taken to be commutative in the following Theorem. Let $\mathcal A$ be a commutative Banach algebra (over ...
0
votes
2answers
19 views

Characteristic and Principal Ideal.

This might be a simple question for some of you, but I am quite confused on the whole concept of principal ideals. Question 1: What is the characteristic of $\mathbb{Z}_2[X,Y]$ where it is the ring ...
1
vote
1answer
33 views

A commutative ring with unity which is not a PIR has a non-trivial ideal generated by two elements which is not a principal ideal?

Let $R$ be a commutative ring with unity which is not a principal ideal ring . Then is it true that $\exists 0\ne x,y \in R$ such that the ideal $\langle x, y \rangle$ is not a principal ideal ?
1
vote
2answers
41 views

How to show that two left ideals over $R$ are not isomorphic $R$-modules?

Let $R=\begin{bmatrix}F&0&0\\ F&F&0\\F&0&F\end{bmatrix}$ with a field $F$. Show that $$\begin{align} ...
2
votes
2answers
32 views

Subset of Principal ideal

Just started learning a little bit about ring theory. Can anyone give me a hint (or counterexample if the statement isn´t true). Let be $P $ Principal ideal then every Ideal $P^{'}\subset P$ is also a ...
0
votes
2answers
42 views

Prime Ideal and Proper Ideal

Could someone please explain to me the definition of a prime ideal and a proper ideal. I honestly do not understand this concept. If possible please explain your version of the definition in the ...
0
votes
1answer
37 views

Prove or Disprove $I=\{a_0x^0+\dots +a_nx^n:a_0+\dots+a_n=0\}$ is a sub ring of field F[x]

Prove or Disprove $I=\{a_0x^0+\dots +a_nx^n:a_0+\dots+a_n=0\}$ is subring of field F[x] We need to show it is a subring that is closed addition closed multiplication (struggling) $\exists $ ...
1
vote
1answer
61 views

Showing that a ring is a field as well for one of the provided choices.

Let $\mathbb M$ be one of the following rings: $\mathbb{R}, \mathbb{Q}, \mathbb{F_9}, \mathbb{C} $. Let $I$ be the ideal generated by $x^4+2x-2$. Is the ring $\mathbb M[x]/I$ a field for some of ...
2
votes
3answers
176 views

Principal ideals in rings.

Consider the following list of principal ideals (2), (3), (5), (6) in the ring ℤ/14ℤ. There are : a) Only one ideal in this list. b) Two distinct ideals in this list. c) Three distinct ideals in ...
0
votes
1answer
20 views

Creating ideals of finitely presented algebras in MAGMA

I want to create (right, left, two-sided) ideals of a finitely-presented algebra in MAGMA. I know how to do this for a free algebra, and the magma handbook indicates that it's possible to do this for ...
5
votes
1answer
53 views

prove $(n)$ prime ideal of $\mathbb{Z}$ iff $n$ is prime or zero

prove $(n)$ prime ideal of $\mathbb{Z}$ iff $n$ is prime or zero Defintions Def of prime Ideal (n) $$ ab\in (n) \implies a\in(n) \vee b\in(n) $$ Def 1] integer n is prime if $n \neq 0,\pm 1 $ ...
1
vote
1answer
50 views

Associated primes of the symbolic power and ordinary power

I am struggling to understand the following quote from a paper of Arsie and Vatne "A note on symbolic powers and ordinary powers of homogeneous ideals": Our interest in the symbolic power stems ...
1
vote
2answers
62 views

A more technical definition of an Affine Variety

My textbook states that an affine variety is Definition. Let $k$ be a field and let $f_1,...,f_s$ be polynomials in $k[x_1,...,x_n]$. Then we set $$V(f_1,...,f_s)=\{(a_1,...,a_n) \in k^n : ...
0
votes
2answers
44 views

Showing that a given subset is an ideal of a polynomial ring

Let $\Bbb Z[x]$ be the set of polynomials in indeterminate $x$ with integer coefficients. Consider the subset $I = \{ 2f(x) + xg(x) | f(x), g(x) ∈ \Bbb Z[x] \}$. Show that $I$ is an ideal in $\Bbb ...
0
votes
2answers
37 views

A commutative ring with unit element, which only has a finite number of ideals, is a field

Let $R$ be a commutative ring with unit element. Suppose that it only has a finite number of ideals. Show that it is a field My reasoning is as follows: Let $a \in R-\{0\}$. Since $R$ is a ...
0
votes
2answers
57 views

Can we give an ring $R$ such that every prime ideal of $R$ be maximal with $|\operatorname{Max}(R)|=\infty?$

It is well known that in commutative rings, maximal ideals are prime. Can we give an example of a ring $R$ such that every prime ideal of $R$ is maximal with $|\operatorname{Max}(R)|=\infty?$
2
votes
1answer
48 views

Is the polynomial ring over a PID also a PID?

As stated in the title, given a principal ideal domain $R$, is the polynomial ring $R[x]$ necessarily a principal ideal domain? In particular, is the polynomial ring $(\mathbb{Z}[i])[x]$ over the ...
1
vote
1answer
40 views

Prove or disprove $(6x,10)=(2x,10)$

Prove or disprove $<6x,10>=<2x,10>$ where they are ideals of $\mathbb Z[x]$. My group teammate said it is, I say it is not $$<6x,10>=\{0,10,6x,6x+10, ...
0
votes
1answer
34 views

Simplify $(x+2+I)(x^2+3+I)$ where $I=<4x,2>$ an Ideal of $ Z[x]$

Simplify (x+2+I)(x^2+3+I) where $I=<4x,2>$ Ideal of $Z[x]$ where $$(x+2+I)(x^2+3+I)=((x+2)(x^2+3)+I)=(x^3+2x^2+3x+6+I) $$ Now I have handwriting that it is a subring. Not 100% that $if ...
1
vote
3answers
60 views

Ideals in a field and relation to homomorphisms.

For a quiz coming up, my professor has asked the class to "know what the ideals in a field look like, and what this tells you about homomorphisms from a field to other rings..." I have looked in the ...
3
votes
0answers
38 views

Proof verification: $\langle 2, x \rangle$ is a prime, not principal ideal

I want to prove that $\langle 2, x \rangle$ is a prime, but not principal, ideal of $\mathbb{Z}[X]$. $\langle 2, x \rangle$ is prime: Suppose $f(x) = f_n x^n + \ldots + f_1 x + f_0$ and $g(x) = g_m ...
1
vote
2answers
32 views

Prime Ideal that isn't Principal

I'm trying to see how maximal ideals, prime ideals, and principal ideals are related for an arbitrary domain $D$. I know that every maximal ideal is a prime ideal. I know that principal ideals are not ...
1
vote
2answers
15 views

Struggling to understand example of Ideal which is not finitely generated

I'm working through an algebraic number theory book, but I can't understand the example shown below: I follow the example up till it assumes that $\frac{p_1}{q_1},...,\frac{p_n}{q_n}$ are the ...
5
votes
1answer
105 views

Idea behind the definition of different ideal

Let $L/K$ be an extension of number fields. Let $I$ be a fractional ideal in $L$ and $$I^*:=\{x\in L \mid \text{Tr}_{L/K}(xI)\subset \mathcal{O}_K\}.$$ The different of $I$ is the following fractional ...
0
votes
0answers
18 views

The sum of two ideals of polynomials

If $I = \{x^3 + 1\}$ and $J = \{x^3 - 1\}$, who is $I + J$? I tried like this: $I + J = \langle I \cup J \rangle = \langle x^3+1, x^3 - 1 \rangle = \dots$ ? Thank you.
4
votes
3answers
61 views

Number of Distinct ideals of $Z_{60}?$

Tried to count all prime numbers between $0$ to $60$ and adding $(0)$ and (R) to it. that is total $19$ , but I saw that answer is $18$. So, Please explain.
4
votes
1answer
75 views

Multi-pullbacks and the relative chinese remainder theorem

Let $I,J$ be two-sided ideals of a ring $R$. In this question I asked for an "automatic" proof of the fact the natural map $R/(I\cap J)\rightarrow R/I\times _{R/(I+J)} R/J$ is an isomorphism (a direct ...
1
vote
1answer
34 views

Canonical map $R/(I\cap J)\rightarrow R/I\times _{R/(I+J)} R/J$ is an isomorphism

From this MSE question I understand the canonical map $R/(I\cap J)\rightarrow R/I\times _{R/(I+J)} R/J$ is an isomorphism for $R$ a commutative ring and $I,J$ ideals. I tried proving this directly ...
0
votes
0answers
44 views

How to get the variety of a 3-dimensional ideal?

I have managed to calculate a Groebner Basis for the problem described here with respect to degree inverse lexicographic term order with help of SINGULAR. Please open the linked page in another tab! ...