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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Two problems in ideal and radical

Let $R$ be a commutative ring with multiplicative identity. Let $I$ be an ideal of $R$. Let $S=\{r \in R: r^n \in I\mbox{ for some natural number }n\}$. Show that $S$ is an ideal of $R$. Give an ...
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Radical of ideals in local one dimensional rings

Let $R$ be a local one dimensional ring. I want to show that for all $ a,b\in R$, $\sqrt{Ra+Rb}$ is equal to $\sqrt{Ry}$ for some $y\in Ra+Rb$ or is equal to $R$.
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Why is this J(all the linear combinations of two polynoimials in F[x]) an ideal of polynomial field F[x]?

Suppose $a(x)$ and $b(x)$ are two non-zero polynomials in the polynomial field $F[x]$ have a gcd $d(x)$ can be expressed as a "linear combination": $$ d(x) = r(x)a(x) + s(x)b(x) $$ where $r(x)$ and ...
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Extended ideals in power series ring

Let $A$ be a commutative ring with $1$ and consider the ring of formal power series $A[[X]]$. If $I \subseteq A$ is an ideal, let $I[[X]]$ denote the set of power series with coefficients in $I$. This ...
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Principal ideals and UFD's

Problems 1-6 form a project designed to prove that if R is a UFD and every nonzero prime ideal of R is maixmal, then R is a PID. Let I be an ideal of R, since {0} is principal, we can assume that $I ...
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If $l(B,A)$ is a prime ideal then $B$ is maximal in $A$

Let $B\subset A$ be commutative rings with identity. Furthermore $B$ is a domain. We are given the set $$l(B,A) = \{ b\in B\setminus \{0\}: B[b^{-1}] = A[b^{-1}]\} \cup \{0\},$$ where $B[b^{-1}]$ ...
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How to understand ideals in $F$, which is a finite commutative ring with $1$

I do not fully understand about ideals in finite rings, and I have to choose the correct answer to the following: If $F$ is a finite commutative ring with $1,$ then (i) Each prime ideal is a maximal ...
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Preimage of a maximal ideal.

My textbook says that if $f: R \rightarrow S$ is a ring homomorphism, where R and S are commutative; then if P is a maximal ideal of S, it might not necessarily be a maximal ideal of R. A ...
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A question on the symbolic powers of a prime ideal

In I. Swanson's notes about primary decomposition the author wrote: The smallest $P$-primary ideal containing $P^n$ is called the $n$th symbolic power of $P$, where $P$ here is a prime ideal of a ...
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Constructing a primary decomposition for principal ideals in a Krull domain

Let $A$ be Krull ring and $\left\{R_l\right\}_{l \in L}$ a defining family of DVRs. Let $a \in A, a \neq 0$. Then $a R_l \neq R_l$ only for a finite number of indices $l$ (otherwise the image of $a$ ...
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Ideals in Gaussian integers

Let $R:=\mathbb{Z}[i]$. Prove that every nonzero prime ideal $\mathfrak{P}$ of $R$ belongs to one of the following families: 1) $\mathfrak{P}=(1+i)R$ 2) $\mathfrak{P}=(a+bi)R$ where ...
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How do we find prime ideals of a ring of integers of a number fileld?

For example for $F=Q(\sqrt{-5})$. the ring of integers of $F =Z[\sqrt{-5}]$.(since $-5\equiv3 \pmod 4$) but how can we determine prime ideals of this? and another problem is the corresponding ...
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Describe ideals of a ring

I have difficulties with a task in linear algebra. R is a ring and R = {a, b, c, d} These are the tables for + and . in R: ...
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Conditions for a ring to be isomorphic to the product of rings.

Let $R,R_1,\dots,R_n$ be rings. Show that $R\cong R_1 \times \cdots\times R_n$ if and only if there exist ideals $I_1,\dots,I_n$ of $R$ such that (a) $I_i\cong R_i$ for all $i$ (b) $R= I_1 ...
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Subrings and homomorphisms of unitary rings

Let $(R,+,\cdot)$ be a ring (in the definition i use multiplication is associative operation and it's not assumed that there is unity in the ring). I've seen two definitons of subring. 1) non-empty ...
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Classification of nonzero prime ideals of $\mathbb{Z}[i]$

I know the classification of Gaussian primes: let $u$ be a unit of $\mathbb{Z}[i]$. Then the following are all Gaussian primes: 1) $u(1+i)$ 2) $u(a+ib)$ where $a^2+b^2=p$ for some prime number p ...
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In the lattice of ideals, what are the lowerbounds of the prime ideals?

Take the lattice of ideals in a non-commutative ring with 1. It is well known that the lowerbounds of all the maximal ideals are the superfluous ideals. Is there a similar characterization for the ...
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What's the motivation of the ideal? [duplicate]

I'm reading a book on Algebra, it introduces the concept of ring after some examples, the concept of ideal. Definition I.1.8. Let $(A,+,\cdot)$ be a ring and $I$ a non-empty subset of $A$. We say ...
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Proof of the uniqueness of maximal ideal

Let $R$ be a commutative ring with $1$. Let $M$ be a maximal ideal of $R$ such that $M^2 = 0$. Prove that $M$ is the only maximal ideal of $R$.
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How to calculate the norm of an ideal?

Would someone please help explain how to calculate the norm of an ideal? I can't find a source that explains this clearly. For example, I know that the norm ...
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Prove that ideal generated by… Is a monomial ideal

Similar questions have come up on the last few past exam papers and I don't know how to solve it. Any help would be greatly appreciated.. Prove that the ideal of $\mathbb{Q}[X,Y]$ generated by ...
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Why do we study prime ideals?

I hope this isn't an inappropriate question here! I'd like to ask the following (perhaps slightly ill-posed) question: why do we study prime ideals in general (commutative or non-commutative) rings? ...
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A noetherian ring whose ideals are idempotent is artinian

I have to prove the folowing: If $R$ is a Noetherian ring, and for every ideal $I$ of $R$ we have $I = I^{2}$, then $R$ is Artinian. My first thought was to try to prove that the nilradical of ...
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How to check that given polynomials form a Groebner basis

I am wondering if some polynomials are given, how do we know whether they form Groebner basis or not. Note that it is not necessary that given poly's form a reduced Groebner basis. I know how to find ...
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radical of sum of two ideals

$I$ and $J$ are ideals in $k[x_1,\cdots,x_n]$. Show that $\sqrt{I+J}=\sqrt{\sqrt{I}+\sqrt{J}}$. I have no idea how to prove it. Can someone help?
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Definition of “invariant in a module”

What does it mean if someone say that the class of an ideal $I$ in a ring $R$ is an invariant of a module $M$?
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What is the quotient $\mathbb Z[\sqrt{3}]/(1+2\sqrt{3})$?

I am currently doing a past paper and it asks the following: Prove that for $I=(1+2\sqrt{3})$ we have $\mathbb Z[\sqrt{3}]/I$ a field with $11$ elements. If I assume standard algebraic number ...
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Question regarding nilpotent ideals of a ring.

I am working on the following: An ideal $N$ is called nilpotent if $N^n$ is the zero ideal for some $n\geq1$. Prove that the ideal $p\mathbb{Z}/p^m\mathbb{Z}$ is a nilpotent ideal in the ring ...
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Zero-dimensional ideals in polynomial rings

I have the following past exam paper question, a similar sort of question seems to come up every year. And I'm completely lost with it... Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated ...
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Is the inverse of a fractional ideal still fractional?

Let $R$ be a Dedekind domain, $K$ the field of fractions of $R$, $\mathfrak{m}$ be a fractional ideal of $R$, i.e. a non-zero finitely generated $R$-submodule of $K$. We define ...
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Norm and square of the ideal $(2,1+\sqrt{-5})$ in the ring of integers

Let $I=(2,1+\sqrt{-5})$ be an ideal of the ring of integers of $\mathbb Q(\sqrt{-5})$. What is its norm $N(I)$? And is $I^2$ principal? My notes say: $1$, $\sqrt{-5}$ is a $\mathbb Z$-basis ...
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Prime ideal and nilpotent elements

If $\mathfrak p \subset R$ is a prime ideal, prove that for every nilpotent $r \in R$ it follows that $r \in \mathfrak p$. The only hint that my tutor gave me was to use induction. Can someone ...
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Maximal ideals in the ring of real functions on $[0,1]$

Assume $S$ to be all continuous functions from $[0,1]$ to $\mathbb R$. How to prove that all maximal ideals of $S$ have the form $M_{x_0}=\{f\in S \mid f(x_0)=0\}$? Thanks in advance.
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Is the ideal $(X^2-3)$ proper in $\mathbb{F}[[X]]$?

Let $\mathbb{F}$ be a field and $R=\mathbb{F}[[X]]$ be the ring of formal power series over $\mathbb{F}$. Is the ideal $(X^2-3)$ proper in $R$? Does the answer depend upon $\mathbb{F}$? Clearly ...
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Finiteness of ideal of given norm

I'm trying to prove that there are only finitely many ideals of any given norm in the ring of integers $\mathcal{O}_k$ over a numberfield $K$. I know there are "standard proofs" (eg How many elements ...
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Ideal generated by a set in a commutative ring without unity

In a commutative ring with unity $1$, call it $R$, the the ideal generated by the set $S=\{a_1,...,a_n\}$ is the smallest ideal of $R$ containing $S$. It can be proven that this ideal is $$ ...
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The set of zero divisors is the union of radicals of annihilators

I am trying to figure out why the statement $$\text{the set of zero divisors }=\bigcup_{0\ne x\in R} \sqrt{\text{Ann}(x)}$$ is true. Here $R$ is a commutative ring, $\text{Ann}(x)=\{r\in R\mid rx=0\}$ ...
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If a polynomial ideal can be generated by $k$ elements, can it be generated by $k$ elements of any generating set?

Let $I = (p_1,\ldots, p_k) \subset \mathbb{C}[x_1,\ldots,x_n]$. If we have a set of $k'$ polynomials such that $(q_1,\ldots,q_{k'}) = I$, can we always find a $k$-member subset such that ...
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Principal Ideal Domain

Let $D$ be a principal ideal domain and let a be some fixed element of $D$. Let $(a)$ denote the ideal generated by $a$. Prove that if $a$ is irreducible and $I$ is an ideal of $D$ such that ...
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How to show an ideal is zero-dimensional? [duplicate]

Let $J$ denote the ideal in $\mathbb{Q}[x,y,z]$ generated by $\{y^2-xy-2xz,y^3+z^2+1, x^2yz-yz\}$. Show that $J$ is zero-dimensional. How do I go about showing this?
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Norm of ideals in quadratic number fields

I do not really understand how to factor ideals in a quadratic field $K = \mathbb{Q}(\sqrt{d})$, mainly because I have some trouble computing the norm of ideals. I think I understand what is going on ...
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Krull Dimension

I'm studying Krull dimension and I'm confused about the definition of $\text{ht}(P)$, which is as I understand is the following: let $$P_0\subset P_1\subset\dots\subset P_n=P$$ be a chain of prime ...
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Union of Associated Primes being finite.

Let $R$ be a commutative Noetherian ring with unit. Let $I=(x_1,x_2,\dots,x_t)$ be a nonzero ideal of $R$. Define $I_n=(x_1^n, x_2^n,...,x_t^n)$. Are there known results about $\cup_n ...
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Help with proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field.

I have been having a lot of trouble teaching myself rings, so much so that even "simple" proofs are really difficult for me. I think I am finally starting to get it, but just to be sure could some one ...
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Help with proof that $I = \langle 2 + 2i \rangle$ is not a prime ideal of $Z[i]$

(Note: $Z[i] = \{a + bi\ |\ a,b\in Z \}$) This is what I have so far. Proof: If $I$ is a prime ideal of $Z[i]$ then $Z[i]/I$ must also be an integral domain. Now (I think this next step is right, ...
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Proving a factorization of ideals in a Dedekind Domain

Let $R=\mathbb{Z}[\sqrt{-13}]$. Let $p$ be a prime integer, $p\neq 2,13$ and suppose that $p$ divides an integer of the form $a^2+13b^2$, where $a$ and $b$ are in $\mathbb{Z}$ and are coprime. Let ...
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$(ax)(ay) = a(xy) \in (a)$

Lemma 26.1 Let $R$ be a commutative ring with unity element $e$. The set $(a) = \{ar : r \in R\}$ is an ideal of $R$. Proof. First, we will show that $(a)$ is a subring of $R.$ Since $a = ae$ then $a ...
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For a ring R, and ideals $A$, $B$, then $AB=A \cap B$ if $A + B = R$

$AB \subseteq A \cap B$ is clear. I have seen reverse inclusion proven thus, Let $x \in A\cap B$. Since $A+B=R$, there exist $a \in A$, $b \in B$, such that $a+b=1$. Then $x= axa + axb + bxa + bxb$. ...
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Factoring the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$

I am trying to factor the ideal $(8)$ into a product of prime ideals in $\mathbb{Q}(\sqrt{-7})$. I am not exactly sure how to go about doing this, and I feel I am missing some theory in the ...
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Solving $x^2+19=y^5$

I was given several exercises and there is a particular one, I am not able to solve. Let it be given that $Pic(\mathbb{Z}[\sqrt{−19}])$ is a finite group of order $3$. Use this to find all integral ...