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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The quotient of a ring by the annihilator of an ideal

Let $R$ be a commutative ring with identity and $I$ an ideal of $R$. It's 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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51 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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Let I and J be ideals of a ring R. Prove that the union of I and J is an ideal of R iff I is a subset of J or J is a subset of I. [duplicate]

I'd like a proof of: Let I and J be ideals of a ring R. Prove that the union of I and J is an ideal of R iff I is a subset of J or J is a subset of I. This is my report no. 3 in my subject ...
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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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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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30 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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70 views

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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Graphs associated with rings and modules

There are several articles in the literature that deals with some interesting graphs associated with rings and modules. For example The zero-divisor graphs D. F. Anderson, P. S. Livingston, The ...
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Stone representation theorem and right(or left-) one-sided ideals in a ring

Consider Marshall Stone's representation theorem: https://en.wikipedia.org/wiki/Stone's_representation_theorem_for_Boolean_algebras I would like to know in whichspecific way, if any, it is connected ...
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56 views

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 $f\in S$ be a nonzerodivisor of $S$. Suppose that the ideal generated by the coefficients of $f$ is $R$. How to ...
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37 views

Radical of An Ideal In $Z_n$ [on hold]

I am searching for an answer in $Z_n$ regarding the radical of an ideal. Consider an ideal $(a)$ in $Z_n$. Can we calculate radical of $(a)$ in general?
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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 $$ ...
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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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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 ...
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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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29 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 ...
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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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41 views

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 $k[y_1,...,y_n]$. ...
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60 views

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 ...
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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 ...
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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. ...
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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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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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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 ...
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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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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 ...
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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 ...
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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} ...
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1answer
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Using Koszul complex [closed]

Let $A$ be a Noetherian local ring of dimension $t$ with maximal ideal $\mathfrak{m}$. If $J\subset A$ is an $\mathfrak{m}$-primary ideal then we have the following complex for $n\in \Bbb N$: ...
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show set is prime ideal

Let I = { (a,0): a E Z} A)show that I is a prime ideal of Z X Z B) by considering (ZXZ)/I , or otherwise , determine whether I is a maximal ideal of ZXZ. (0,0) is in I so I is non-empty let (a,0) ...
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23 views

Principal ideals containing an ideal in a Noetherian integral domain

Let $R$ be a Noetherian integral domain and $I$ a nonzero ideal consisting only of zero divisors on $R/(x)$, where $x$ is a nonzero element of $I$. Could we always find an element $y\notin (x)$ such ...
3
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84 views

Prove $I$ is non-principal ideal of $\mathbb{Z}[x]$? [duplicate]

I'm new to algebra and got stuck with concept of ideals. The question is to prove that $$I = \left\{ {{a_0} + {a_1}x + \cdots + {a_n}{x^n} \mid {a_i} \in \mathbb{Z},{a_0} \in 2\mathbb{Z}} ...
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Length of a ring? Lenth of a (right or left) ideal

I have seen the concept of length being applied to rings. What is exactly mean by that? What does length mean in a statement like "the composition length of RR is 2, but the composition length of RR ...
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Tensor products of ideals

Consider the class of complex algebras where the tensor products are over complex numbers. Given a complex algebra $A$ and a left ideal $L$ of $A$ generated by $n$ elements. Is $L^{\otimes n}$ ...
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Lasker-Noether Theorem and Kummer-Dedekind

I would like to know about the relations between Ernst Kummer's invention of complex ideal numbers (and Dedekind's development of them into what is now called ideals) regarding the unique ...
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A question regarding one-sided right and left ideals (their calculation for matrix rings)

I have a question regarding the calculation of right and left ideals of a matrix ring. I understand the concept, or so I believe, but, when consulting the argument on this page ...