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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An exercise about field automorphisms and ideals.

Consider a field $K$ and the $K$-algebra $K[x_1,\ldots,x_n]$ of polynomials in $n$ variables; $\mathfrak a$ is an ideal of $K[x_1,\ldots,x_n]$ and suppose that there exists a field $L\subseteq K$ ...
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Finding a maximal ideal in the set of continuous real-valued functions on $\mathbb{R}$ [duplicate]

Let $X$ be the set of all continuous and real-valued functions on $\mathbb{R}$. X is a commutative ring with pointwise addition and multiplication. Let $\alpha \in \mathbb{R}$ be arbitrary. ...
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Proof that an ideal $M$ is maximal iff $R/M$ is a field

I am referencing the proof located at http://www.maths.nuigalway.ie/MA416/section3-4.pdf, Theorem 3.4.2. I am only looking at the right to left direction. I understand the following: Let $a \in I, a ...
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Finding a polynomial that generates an ideal of a polynomial ring

Let $a,b \in \mathbb{R}[x], a = x^5 - x^3 + 2x^2 - x, b = x^5 - x^4 - 8x + 5$. Let $I$ be the ideal in $\mathbb{R}[x]$ generated by a and b. Find a polynomial $p$, with $p \in \mathbb{R}[x]$ and $I ...
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In $\Bbb Z$, what element generates the ideal $(4,7)$?

I have a really silly question. $\mathbb{Z},+,\cdot$ is a HID, so all ideals are principal ideals. Now, $(4,7)$ is an ideal in $\mathbb{Z}$, so it must be a principal ideal, but which element is its ...
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$R$ local ring, $I$ maximal ideal then $x\notin I$ implies $x$ unit

Let $R$ be a conmutative local ring, $I$ its maximal ideal. I want to prove that $x\notin I$ implies $x$ unit. So far I have: Let $x\notin I$, I consider $x+I\in A/I$, which is a field (because $I$ ...
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$I = (x^2, y^2) ⊂ K[x, y]$; $gin\ (I)=?$

an easy Google search give a lot of results about the definition of generic initial ideal. But all definitions I see, are like this one: I can't use this definition to compute gin(I) even in simple ...
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About fractional ideals in dedekind domain

Suppose $I$ and $J$ are two nonzero fractional ideals in the Dedekind domain $R$ and that $I^n = J^n$ for some $n\neq0$ . Prove that $I = J$. We have known every fractional ideal is invertible in ...
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How many elements does $\mathbb Z_7[i]/\langle i+1\rangle$ have?

How many elements have $\mathbb Z_7[i]/\langle i+1\rangle$ ? Elements of $Z_7[i]$ are of the form $a+bi$ $i+1$ is considered as zero in the quotient; $i+1=0\iff i=-1\iff -1=i^2=1$ does it not ...
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What's wrong with this proof that all UFDs are Bezout?

First, some context. I am working with Dummit and Foote's Abstract Algebra, 2nd edition. I stumbled upon this while working on Section 8.3 Exercise 11, which is to prove that all Bezout UFDs are PIDs. ...
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betti-numbers of Gin(I), generic initial ideal of $I$

here in the paper Ideals with Stable Betti Numbers there is a theorem that I can't uderstand it, both in details (which highlighted) and sketch of the proof of (b): can you help please?
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Standard name for ideals generated by a subset of indeterminates?

I have been working on a problem in the polynomial ring $k[x_1,\ldots,x_n]$, where I've been dealing with ideals generated by subsets of the indeterminates, i.e., ideals of the form $$\langle x_i\mid ...
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Generalisation of chinese remainder theorem on ideals of ring without 1

Let $I_1,\dots,I_n$ be (two-sided) ideals of a ring $R$ (not necessarily with 1), which are pairwise co-maximal, i.e. $\forall i\ne j\in \mathbb{Z}_{[1,n]}$, $I_i+I_j=R$. Let $f:R\to R/I_1\times ...
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Krull dimension of the quotient by a single element

Let $(R,m)$ be a Noetherian local ring and let $M$ be a finitely generated $R$-module of dimension $d$. The Krull dimension of $M$ is defined to be the Krull dimension of $R/\operatorname{ann}(M)$. ...
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Relation between lattice theorem in groups and in rings

I was studying my abstract algebra notes, and couldn't help but notice a striking similarity between the following two statements: Let $G$ be a group, and $H\triangleleft G$. The canonical ...
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If $J$ is the ideal generated by all idempotents in a prime ideal, then $R/J$ has only trivial idempotents

Let $R$ be a commutative ring with identity, $P$ be a prime ideal in $R$ and define $$X := \lbrace t \in P \mid t^2=t \rbrace. $$ Also let $J$ denote the smallest ideal of $R$ that contains $X$. ...
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How can the only maximal ideal of $C[x] / X^2$ be $(X)$?

In my notes I have the following example which I don't understand. Let $f$ be the canonical injection from $C$ to $C[X]/X^2$.The only maximal ideal of $C[X]/X^2$ is $(X)$ and $f^{-1}((X))$=$(0)$. ...
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Do surjective ring homomorphisms commute with intersection of ideals?

Let $f:A\longrightarrow B$ be a surjective ring homomorphism. Is it true that for any intersection of ideals, the image of the intersection is equal to the intersection of the images of the ideals? ...
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Property of ideal [closed]

Let $R$ be an associative algebra. Let $I$ be an ideal of $R$. Let $J$ be an ideal of the algebra $I$. Prove that $(J)_R$ the ideal of $R$ generated by elements $J$, has the property: ...
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Why is $I + (a) = R$?

Let $I$ be a maximal ideal of a ring $R$ and let $(a)$ be the principal ideal generated by the element $a$ which lies in $R$ but not $I$. Why does $R = I + (a)$? Thanks in advance
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showing an easy set is an ideal

I'm having troubles understanding the definition of an ideal. I found this example online, but the author did not explain the steps and just stated it was an ideal. I hope someone could show me the ...
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Equivalent definitions of homogeneous ideal

I need to show that an ideal $I$ of a $\mathbb{Z}$-graded ring R is homogeneous iff for every element $f \in I$, all homogeneous components of $f$ are in $I$. $\Leftarrow$ implication is obviously. ...
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Show the quotient ring R/I is not a field

Studying for an exam in Algebra. Let $R=\mathbb{Z}[i]$ with the usual normfuction $N, z = 5+3i$ and $I = \, <z>$ Show that z isn't a prime element in $R$ and that $R/I$ isn't a field. I ...
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$IJ$ is the set of nilpotent elements

Let $R$ be a commutative ring with identity which is Noetherian. Let $V(A)$ denote the set of all prime ideals of $R$ containing the ideal $A$. Suppose that $V(0) = V(I) \cup V(J)$ and $V(I) \cap V(J) ...
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$0$ and $p^n \mathbb{Z}_p$ are the only ideals of $\mathbb{Z}_p$

I am looking at the proof of the sentence: $\mathbb{Z}_p$ contains only the ideals $0$ and $p^n \mathbb{Z}_p$ for $n \in \mathbb{N}_0$. It holds that $\bigcap_{n \in \mathbb{N}_0 p^n \mathbb{Z}_p}=0$ ...
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Is the complement of a prime ideal closed under both addition and multiplication?

Let $P$ be a prime ideal in a commutative ring $R$ and let $S=R-P$ ,i.e. $S$ is the complement of $P$ in $R$. Then, justify with reason which of the following(s) are correct: $S$ is closed under ...
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Is $\{x f(x)+3g(x) \;|\;f,g\in \mathbb{Q}[x]\}$ a (main) ideal?

Is it possible to show whether or not $ \{xf(x)+3g(x)\;|\;f(x),g(x) \in \mathbb{Q}[x]\} $ is an ideal (or main ideal in $\mathbb{Q}[x]$)? I know how to prove it for $\mathbb{Z}[x]$, but what with ...
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What is the definition of $I=(f(X,Y),g(X,Y))$?

What is the definition of this ideal in $\mathbb C[X,Y]\ I=(f(X,Y),g(X,Y))$ for some polynomials $f,g \in \mathbb C[X,Y]$
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For which countable successor ordinals $\alpha$ is the reverse order isomorphic to the ideals of a PID ordered by inclusion?

Let $\alpha$ be a countable successor ordinal and $\alpha^{\mathrm{op}}$ the reverse order. For which $\alpha$ is there a commutative principal ideal ring $R$ such that the ideals of $R$ form a chain ...
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Spectrum of $\mathbb R[X,Y]$ [duplicate]

Let $A=\mathbb R[X,Y]$. Is it easy to classify the $\operatorname{Spec}A$? I guess it contains at least $(0)$ and $(p)$ for primes $p\in A$ but maybe some else sets. Is it easy to classify those? ...
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A minimal prime ideal consists of zerodivisors [duplicate]

Let $A$ be a unital commutative ring (I do not assume $A$ to be Noetherian). Let $\mathfrak{p} \subset A$ minimal prime ideal. Question: Are all elements of $\mathfrak{p}$ zero divisors? Comment: I ...
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Equivalent conditions for an ideal to be prime

Let $R$ be a commutative ring. An ideal $I$ is called prime if whenever $ab\in I$ then $a\in I$ or $b\in I$. I want to show that $I$ is prime if whenever $JK\subseteq I$, then $J\subseteq I$ or ...
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Does $(X)(Y)=(XY)$ for $X,Y\subseteq R$?

Let $R$ be a commutative ring. Denote by $X\ast Y=\{xy\mid x\in X,y\in Y\}$ the complex product of subsets. I want to show that given subsets $X,Y\subseteq R$ the following ideals are equal: ...
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Ideals of $\mathbb{Z}/{p_{1}^{k_{1}}..p_{m}^{k_{m}}}$

I need to describe all the ideals of $\mathbb{Z}/{p_{1}^{k_{1}}..p_{m}^{k_{m}}}$ I suppose that trivials, and $(0,..,1_{i},..,0)\mathbb{Z}/{p_{1}^{k_{1}}..p_{m}^{k_{m}}}$ for any $i$ and nilradicals ...
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$p^2=p$ in closure of ideal $I$ of Banach algebra implies $p\in I$.

Let $I\subset A$ be a ideal of a Banach algebra $A$. Assume $p\in \overline I$ and $p^2=p$. Show: $p\in I$. Can someone give me a little hint how to solve this, please?
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Height and coheight of an ideal

Given an ideal $\mathfrak{a}$, Matsumura defined the height of $\mathfrak{a}$ as: $$\text{ht}(\mathfrak{a})=\inf_{\mathfrak{p}\in V(\mathfrak{a})}\text{ht}(\mathfrak{p})$$ He states that: ...
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Set of integer p-adics-Proposition

Proposition: "$\mathbb{Z}_p$ contains only the ideals $0$ and $p^n \mathbb{Z}_p$ for $n \in \mathbb{N}_0$. It holds $\bigcap_{n \in \mathbb{N}_0} p^n \mathbb{Z}_p=0$ and $\mathbb{Z}_p \ ...
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The ring is a principal ideal domain, especially an integral domain.

The following holds for the ring $ \mathbb{Z}_p, p \in \mathbb{P}$: The ring $ \mathbb{Z}_p $ is a principal ideal domain, especially an integral domain. I try to understand the following proof: ...
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Prove that an ideal is not maximal

Ring $\mathbb Z[x],$ ideal is $(x)$. How to prove that this is NOT a maximal ideal? I can't imagine ideal, part of which would be $(x)$.
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Example of ideals such that $I^n=0$ but $I^{n-1}\not= 0$

Let $R$ be a ring. For each $n>0$ I want to find an ideal $I$ of $R$ such that $I^n=0$ but $I^{n-1}\not= 0$. Clearly this won't work for $R=\Bbb{Z}$ or $\Bbb{Z}/n\Bbb{Z}$. And I ran out of ...
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Showing that an epimorphism of an ideal is again an ideal

Let $R, S$ be commutative rings, $f : R \rightarrow S$ an epimorphism, I an ideal of R. Show that $f(I)$ is an ideal of $S$. As far as I understand, I need to show 4 things: 1) $0_s \in f(I)$ ...
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Alternative proof of '$I$ is maximal iff $R/I$ is a field'

For any commutative ring $R$ and an ideal $I$ of $R$, $I \neq R$, show that $I$ is a maximal ideal iff $R/I$ is a field. I write my own proof and it checks with the 'traditional' proof which ...
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Finding the ideal

Determine all the ideals, prime ideals, and maximal ideals of $\mathbb{R}[x]/I$ where $I$ is the ideal generated by $(x^2+1)(x-2)^2$. I am currently doing some reading on ideals (see ...
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Determine whether an ideal is principal or not

Let $I=\{a+b\sqrt{-3}: a+b \text{ even}\}$ be an ideal in $R=\mathbb{Z}[\sqrt{-3}]$. I want to determine whether $I$ is a principal ideal or not. I've been trying to work with the ideal $(2)$. I ...
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maximal ideal problem [duplicate]

I want to solve this problem, but I have no idea how I can start: If $K$ is a field, $(a_1,...,a_n) \in K^n,$ and $I$ the ideal $I=\langle x_1-a_1,...,x_n-a_n\rangle$, then how can we prove that ...
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Why $P_1\neq P_1P_2$?

Question: If $P_1,P_2$ are distinct prime ideals of an artinian ring, why is it that $P_1\neq P_1P_2$? I know that prime ideals of an artinian ring are maximal, but still, I can't see why ...
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The interpretation of ideals of a ring.

Ideals of a commutative ring (I have only studied the commutative case) are thought of as generalized numbers (in algebraic number theory) and as ring homomorphisms (through the ideal as kernel ...
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$I$ and $J$ are coprime ideals iff $x \to (x + I, x + J)$ is surjective.

I'm stuck on this exercise and any help would be well appreciated: Let $R$ be a commutative ring with ideals $I,J$. Show that $R=I+J$ if and only if $\phi(x)= (x + I, x + J)$ is surjective from ...
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50 views

Confused on notions of maximal ideal and some notation

I'm just getting started learning ring theory and am currently learning about ideals. By book (Dummit & Foote) says the following: For example, in the ring $R = \mathbb{Z}[x]$ the elements $2$ ...
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one to one correspondence of Ideals in a ring and its localization

Let $A$ be a commutative ring, and $S$ a mutiplicatively closed subset. In my text book, it is stated that: there is one to one correspondence of prime ideals in ring $A$ (not meeting $S$) and ...