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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Ideal of polynomials vanishing on $\{(x,y): x^2+y^2=1, x \neq 0 \}$

I'm reading the book "Introduction to algebraic geometry" by Hassett, and in Chapter 3, after introducing the concept of the ideal of polynomials vanishing on a set $S$, the author gives some examples,...
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Centre of matrix ring over skew field

Let $R$ be a semisimple ring. Show that $R$ is simple iff the centre of $R$ is a field. Book's solution: If $R$ is simple, it has the form $\mathfrak{M}_n(K)$ for a skew field $K$, and its ...
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Determinant of the change of basis for fractional ideal

Let $A$ be a fractional ideal of some number field extension $K:\Bbb Q$. Let $\omega_1, \dots ,\omega_n$ be a $\Bbb Z$ basis for $\mathcal O_K$ and let $\alpha_1, \dots ,\alpha_n$ be a $\Bbb Z$ basis ...
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How do I prove that primary ideals satisfy this property?

Let $R$ be a commutative ring. Let $Q$ be a primary ideal of $R$. Let $I,J$ be ideals of $R$ such that $IJ\subset Q$. How do I prove that $I\subset Q$ or $J^n\subset Q$ for some positive integer $n$...
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57 views

Looking for an example of an ideal contained in the union of other ideals, but not in any ideal individually

I'm looking for an example of the following scenario: $A, B, C $ are three ideals such that $C\subseteq A\cup B $ but $C\not\subseteq A $ and $C\not\subseteq B$. Any help would be great!
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If $R$ is a simple Artinian ring, then when is a finitely generated module free?

Here's an exercise from my book, which only gives a brief solution which leaves me very confused. Let $R$ be a simple Artinian ring, say $R=K_r$. Show that there is only one simple right $R$-...
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If a set $S$ generates an ideal $I\subset F[x_1,x_2,\ldots,x_n]$, then there is a finite subset $S_0 \subseteq S$ which generates $I$

The question: If $I$ is an ideal in $F[x_1,x_2,\ldots,x_n]$ generated by a set of polynomials $S$, then there is a finite subset $S_0 \subseteq S$ which generates $I$. By the Hilbert Basis ...
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Let $\phi\in \mathbb{C}[V]$. Show that $\mathbf{V}_V(\phi)=\emptyset$ if and only if $\phi$ is invertible in $\mathbb{C}[V]$.

This is an exercises in Ideals, Varieties and Algorithms by Cox et al. Let $V\subset \mathbb{C}^n$ be a nonempty variety. Let $\phi\in \mathbb{C}[V]$. Show that $\mathbf{V}_V(\phi)=\emptyset$ if ...
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Is this another way of stating the Chinese Remainder Theorem?

Assume that $I + J = R$. Let $a,b \in R$. Find an element $u$ of $R$ satisfying $a + I = u + I$ and $b + J = u + J.$ I want to work on this, but I feel there's some issue of a missing theorem I don'...
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Is a polynomial $f$ zero at $(a_1,\ldots,a_n)$ iff $f$ lies in the ideal $(X-a_1,\ldots,X-a_n)$?

This is probably a very silly question: If $R$ is an arbitrary commutative ring with unit and $f\in R[X]$ a polynomial, then for any element $a\in R$ we have $$f(a)=0 \Longleftrightarrow X-a ~\mbox{ ...
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111 views

The quotient of a ring by the annihilator of an ideal

Let $R$ be a commutative ring with identity, and $I$ an ideal of $R$. Is it true that we have an $R$-module isomorphism $$I\cong R/ann_RI,$$ where $ann_RI=\{x\in R:xr=0,\;for\;all\;r\in I\}$ is the ...
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63 views

Ideals and injective modules

Let $I$ be a left ideal of $R$. Assume that there exist element in $I$, which is not a zero divisor. How to prove that for every (left) injective $R$-module $Q$ we have $IQ=Q$ ? I need only hints.
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81 views

Every modular right ideal is contained in a modular maximal ideal

If $R$ is a ring, possibly without $1$, a right ideal $\mathfrak{a}$ of $R$ is modular if there exists $e\in R$ such that $r-er\in \mathfrak{a}$ for all $r\in R$. So $e$ is a left identity mod $\...
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Looking for a terminology in ring theory (“ideal” which is not necessarily closed under addition )

I am wondering if there is a name for the subsets $S$ of a commutative ring $R$ such that for every $r\in R$ and every $s\in S$ we have $rs\in S$. Thus $S$ is ...
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68 views

What is the difference between submodules of $A/\mathfrak a$ as $A$-module or as $A/\mathfrak a$-module?

If $\mathfrak a$ is an ideal of unital commutative ring $A$, then we can consider $A/\mathfrak a$ as $A$ module or as $A/\mathfrak a$ module. If $A=\mathbb Z$ there is no structural difference between ...
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Prime ideal $P$ of $\mathbb{Z}[x]$ such that $P \cap \mathbb{Z}=\{0\}$ is principal

The problem stated more precisely is this: Let $P$ be a prime ideal of $\mathbb{Z}[x]$ such that $P \cap \mathbb{Z} =\{0\}$. Show that $P$ is a principal ideal. I think there is a problem with my ...
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53 views

Example of a Non-Graded Ideal in a Graded Ring

A ring $S$ is said to be graded if there are additive subgroups $S_0, S_1, S_2, \ldots$ such that $S=\bigoplus_{k\geq 0}S_k$ and $S_iS_j\subseteq S_{i+j}$ for all $i$ and $j$. An ideal $I$ in a ...
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If $R$ is a commutative ring, the nilpotents form necessarily an ideal of $R$? [duplicate]

This is an algebra question from an exam a few years ago: Let $R$ be a ring, and let $N = \{a \in R: a^n = 0 \text{ for some } n \in \mathbb{N}, (n \text{ depends on } a) \}$. Prove or disprove: ...
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22 views

Stone representation theorem and right(or left-) one-sided ideals in a ring

Consider Marshall Stone's representation theorem: I would like to know in which specific way, if any, it is connected with the determination of right or left one-sided ideals in a ring. Simple ...
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Prove that $f$ is a nonzerodivisor on $R[x_1,\dots,x_r]/IR[x_1,\dots,x_r]$ for every ideal $I$ in $R$

Let $R$ be a Noetherian commutative ring with unity, and $S=R[x_1,\dots,x_r]$. Let $f\in S$ and suppose that the ideal generated by the coefficients of $f$ is $R$. How to show that $f$ is a ...
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Uniqueness of the decomposition of an ideal

Let $ F $ be a non-empty subset of $ \{ 1,2,\dots,n\} $ and $ P_{F}=(\{x_{i}:i\in F\}) $. Let $ F_{1},F_{2},\dots,F_{m} $ be pair-wise distinct non-empty subsets of $ \{1,2,...,n\} $ and $$ I=\prod_{...
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Noether & Schmeidler- Hurwitz-Ideals

Consider the following page from Noether and Schmeidler's 1921 work: http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN266833020_0008&DMDID=DMDLOG_0008&LOGID=LOG_0008&PHYSID=PHYS_0013 ...
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Left Ideals and Two-Sided Ideals of Rings of Matrices

This is a question I posted in this topic: Pathologies in "rng". However, I decided that the question deserves its own thread, and I want to know if anybody can answer it. For ...
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71 views

Radical ideal in $\mathbb{R}[x,y,z]$

In $\mathbb{R}[x,y,z]$ is the ideal $I=\left\langle xz,yz\right\rangle$ radical? If $f \in I$ tried write $f=g.xz+h.yz+ax+by+c$ and conclude that $f^m \notin I$, if $m>0$, but I could not.
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Why is $(2,x)$ non-principal in $\Bbb Z[x]$? [duplicate]

Why is $(2,x)$ non-principal in $\Bbb Z[x]$? Apparently this is the case, I just read it on wiki, as a counter example to $\Bbb Z[x]$ being a PID. What is $x$ here? I mean $2$ can surely generate $x$...
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Invertibility of translating ideals from a ring to its localization

Let $A$ be a ring with some multiplicative subset $S$. Define $AS^{-1} = \{\frac as| a \in A, s \in S\}$. Let $I_A$ and $I_S$ be the sets of ideals of $A \subset A-S$ and $AS^{-1}$ respectively. ...
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Are these rings fields?

Are the following rings fields? 1) $\Bbb Q[x] /\langle x^2+1\rangle$ Since a polynomial ring taking values on any field is a E.D, and hence a P.I.D, this is a field iff the ideal is prime or ...
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37 views

Quotient ring is a ring homomorphism

Why is this a ring homomorphism? $$\phi:R\to R/I$$ where $I$ is an ideal, given by $\phi:r\mapsto r+I$. To be a ring homomorphism it needs to be a homomorphism of addition and multiplication, i.e: ...
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Prove that: $B/A \triangleleft R/A$ if $A \subseteq B ;\ \ A, B \triangleleft R $(ring)

Prove that: $B/A \triangleleft R/A$ if $A \subseteq B ;\ \ A, B \triangleleft R $(ring) : $ \triangleleft $ means ideal. I need this proof to continue on, I'm told it's not that hard, but I just can;...
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In the ring of polynomials $F[t]$, every ideal is principal

Let $$\langle a \rangle=\bigcap_{ a \in I} I$$ $$ \langle a \rangle =\{ Ira + na, r\in R,n \in Z\}$$ What is unclear is why$$ I=\langle 0 \rangle$$ proves that I is a principal ideal. My definition ...
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Show that $\alpha_A^{-1}(I'+J')=\alpha_A^{-1}(I')+\alpha_A^{-1}(J')$, where $I',J'$ are ideals and $\alpha_A$ is a surjective ring homomorphism.

Let $\alpha_A: k[x_1,...,x_m]\rightarrow k[y_1,...,y_n]$ be a map defined by $\alpha_A(f)(y)=f(Ay)$ where $A$ is an $m\times n$ constant matrix. Let $I',J'$ be ideals in $k[y_1,...,y_n]$. Show ...
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Determine which of the following rings are fields.

Have I done it correctly? Determine which of the following rings are fields: a) $(\mathbb{Z}/2\mathbb{Z})[x]$/$\large_{(x^2+1)}$ b)$(\mathbb{Z}/3\mathbb{Z})[x]$/$\large_{(x^2+1)}$ My ...
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Show that $\left\langle\alpha_A(I\cap J)\right\rangle \subset \left\langle\alpha_A(I)\right\rangle \cap \left\langle\alpha_A(J)\right\rangle $.

Let $\alpha_A: k[x_1,...,x_m]\rightarrow k[y_1,...,y_n]$ be a map defined by $\alpha_A(f)(y)=f(Ay)$ where $A$ is an $m\times n$ matrix. Show that $\left\langle\alpha_A(I\cap J)\right\rangle \subset \...
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Prove $a\mathbb{Z}[x]+x\mathbb{Z}[x]$ is a principal ideal on $\mathbb{Z}[x] \iff a=0$ or $a=1$ or $-1$.

I've tried several things, but I don't know how to properly show it. Prove $a\mathbb{Z}[x]+x\mathbb{Z}[x]$ is a principal ideal on $\mathbb{Z}[x] \iff a=0$, $a=1$ or $a=-1$. My try: Proof: ...
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Ring localization and ideals

I'm trying to solve a couple of problems involving ring localization and I'm not sure if my solutions are right or if I understand the idea of localization correctly. Let $A$ be a commutative ring,...
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Clarification on notation in Siegfried Bosch's Commutative Algebra book about primary decomposition of ideals.

I'm reading through Siegfried Bosch's Commutative Algebra book, and I'm confused on his notation in one his proofs. He uses this notation a lot, so I think I should I understand it. The notation first ...
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Prove that $I_1^m\cap I_2^m \cap \dots \cap I_r^m=I_1^m\cdots I_r^m$, where $I_1,…,I_r$ are ideals in $k[x_1,…,x_n]$ and are comaximal.

This is an exercise from Ideals, Varieties and Algorithms by Cox, etc. If $I_i$ and $J_i=\cap_{j\ne i}I_j$ are comaximal for all $i$, where $I_1,...,I_r$ are ideals in $k[x_1,...,x_n]$, prove that $...
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When $f(I)S=S$ for each ideal $I$ of $R$?

Let $R$ and $S$ be commutative rings (with $1$) and $f : R\to S$ be a ring homomorphism. For an ideal $I$ of $R$, set $I^e:=\langle f(I)S\rangle$ (called the extension of $I$ to $S$). Question 1. ...
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An alternate definition of ideals

A filter on a poset if by definition its nonempty subset $F$ such that it does not contain the least element and $A, B \in F \Leftrightarrow \exists Z \in F : (Z \le A \wedge Z\le B)$ for every ...
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Localization of ideals at all primes

Let $R$ be a commutative ring with $1$ and $I$, $J$ ideals in $R$. For a prime ideal $P$, let $I_P=(R-P)^{-1}I$ be the localization of $I$ at $P$. Question: If $I_P=J_P$ for all prime ideals $...
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In a reduced ring the set of zero divisors equals the union of minimal prime ideals.

If $R$ is a reduced commutative ring with identity, why is the set $Z$ of zero divisors the union of minimal prime ideals? I know that $Z$ is a union of associated primes, and that the intersection ...
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Express the ideal $(6) \subset\mathbb Z[\sqrt {-5}]$ as a product of prime ideals.

Express the ideal $(6) \subset\mathbb Z[\sqrt {-5}]$ as a product of prime ideals. I know I can write $(6)=(2)(3)=(1+\sqrt {-5})(1-\sqrt {-5})$. But I guess these factors might not be prime. What'...
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If the intersection of ideals $I_{1},\ldots,I_{n}$ is contained in a prime ideal $P$, then one of them is contained in $P$

Let $A$ be a commutative ring and $I_{1},\ldots, I_{n}$ and $P$ ideals in $A$ with $P$ prime so that $\cap_{i=1} ^{n} I_{i} \subset P $. Show that there's an $i_0 \in \{1,...,n \}$ so that $I_{i_0} \...
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1answer
59 views

Height and minimal number of generators of an ideal.

Can anyone could give me a reference in a book about the proof of the following Let $I$ be an ideal of a ring. We denote with $\operatorname{ht}(I)$ the height of $I$, and by $\mu(I)$ the minimal ...
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If $\mathcal{L}$ is a minimal left ideal of an algebra $\mathcal{A}$, then $\mathcal{A}l=\mathcal{L}$ for all $l\in \mathcal{L}$?

I am going through Hassani's 'Mathematical Physics': He defines an algebra $\mathcal{A}$ as a vector space together with a multiplication map $\mathcal{A}\times\mathcal{A}\rightarrow\mathcal{A}$. He ...
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86 views

Showing there are at least one and only finitely many maximal ideals containing the extension of a maximal ideal [closed]

Let $F$ be a field and $M$ a maximal ideal of $F[x_1, x_2, ..., x_n]$. Let $K$ be an algebraic closure of $F$. Show that $M$ is contained in at least one and in only finitely many maximal ideals of $K[...
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generating element of $I = \{p \in \mathbb{Q}[X]: p(0)=0, p'(0)=0\}$

I have the ideal $I = \{p \in \mathbb{Q}[X]: p(0)=0, p'(0)=0\}$. I have verified that it is an ideal by multipyling an arbitrary element of the ideal with an arbitrary element of $\mathbb{Q}[X]$ and ...
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140 views

Prime ideal in $\mathbb Z[\sqrt{10}]$

I am trying to solve this exercise: Prove that $\langle 2,\sqrt{10} \rangle$ is a prime ideal in $\mathbb Z[\sqrt{10}]$. I could do the following: I pick an element of the form $zw \in \langle 2,\...
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1answer
45 views

Can we make sense, in general, of taking a quotient by multiple ideals?

I feel that this is a rather silly question, stemming from a fundamental misunderstanding of quotients, but I'm not quite able to make it precise. My question is: given two ideals $\mathfrak{a,b}\...
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Left ideals in a subring of $M_2(R)$

Let $R$ be a 2-dimensional complete regular local ring $R$ over an algebraically closed field $k$, that is $R\cong k[[x,y]]$. Now look at the the following subring $A$ of $M_2(R)$: $A=\begin{pmatrix} ...