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 factor ring of the n-th power of a maximal ideal is local. [closed]

Let $M$ be a maximal ideal in a commutative ring $R$ with identity and $n$ is a positive integer, then the ring $R/M^n$ has a unique prime ideal and therefore is local. It is easy to see that unique ...
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Why is the generating set a proper ideal of…? [closed]

Why is $\langle 89, 3-4\sqrt{-5}\rangle$ a proper ideal of $\Bbb{Z}[\sqrt{-5}]$?
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Polynomial Ring Divided By Principal Ideal

Let $F_5[X]$ be the polynomial ring over $F_5$ and $I = <X^2+X+1> $. Show that any element of $\frac{F_5[X]}{I}$ can be written as $a+bX+I$ where $a,b$ are in $F_5$. I guess I can be written ...
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Cluesless over a proper ideal question [duplicate]

Let $X=\{ \alpha+\beta\sqrt{-5} \mid \alpha,\beta \text{ are rational numbers} \}$ Does there exist an integer $a$ such that $a$ and $3-4\sqrt{-5}$ generate a proper ideal of $X$? Can anyone answer ...
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Number of generators of a given ideal.

Let $I=\langle 3x+y, 4x+y \rangle \subset \Bbb{R}[x,y]$. Can $I$ be generated by a single polynomial? My approach: If $I$ can be generated by a single polynomial, then the two "apparent" ...
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Showing that $1$ is an element of the ideal $\langle 3,x,n\rangle$ in $\mathbb Z[x]$ if $\operatorname{gcd}(3, n) = 1$ [closed]

Can anyone explain me the following step: If I have the ideal $\langle 3,x,n \rangle$ in $\mathbb Z[x]$ where $\operatorname{gcd}(3,n)=1$, then $1\in \langle 3,x,n \rangle$. Kindly help as I'm ...
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How to prove $(2)=(2x)$ for $x\in \mathbb{Z}$?

How to prove $(2)=(2x)$ for $x\in \mathbb{Z}$? ($(2)$ is the principal ideal generated by $2$ over the ring of integers.) I wrote $(2x)=\{(2x)z:z \in \mathbb{Z}\}=\{2y:y \in \mathbb{Z}\}=(2)$ where ...
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Isomorphism of a Quotient Ring

Show that $(A \times B) \big/ (\mathfrak{a} \times \mathfrak{b}) \cong \big(A \big/ \mathfrak{a} \big) \times \big( B \big/ \mathfrak{b} \big)$. Where $A$ and $B$ are commutative rings. I am ...
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Show that for a field $F$, the polynomial ring $F[x_1, x_2, \ldots, x_n]$ is not a PID for $n>1$.

I want clarification of the following solution: Let $I=(x_1)+(x_2)$ be an ideal of $F[x_1, x_2, \ldots, x_n]$. Then if $I=(f)$ is principal then we must have $f \in F \backslash \{0\}$ since ...
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Determine a parameter so that a given ideal equals $ℝ[x,y]$

Let $$B=\langle 3x+y-m, 4x+y\rangle⊆ ℝ[x,y].$$ Find a parameter $m$ so that $$B=ℝ[x,y].$$ My attempt so far: If $1_{ℝ[x,y]}\in B$ then we get the desired equivalence, since any ideal ...
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Proving maximal ideals when considering an irreducible polynomial

For $F$ a field, and $q(x)\in F[x]$ Suppose that $q(x)$ is a irreducible polynomial within the ring. Prove that $ \langle q(x) \rangle$ is a maximal ideal of $F[x]$ I've already proved that ...
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How to prove that two principal ideals are equal [on hold]

For $F$ a field, and $q(x)$ a polynomial in the polynomial ring $F[X]$, with $a\in F$ where $a \neq 0$, show that $\langle q(x) \rangle= \langle aq(x) \rangle$, where $\langle \cdot \rangle$ denotes ...
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Under what conditions is $J\cdot M$ an $R$-submodule of $M$?

I have that $M$ is an $R$-module where $R$ is commutative and unitary ring. Supposing that $J$ is an ideal of $R$, when is the set $J \cdot M$ an $R$-submodule of $M$? I have to check the two axioms ...
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Why this ideal is a subset of $K[x_0, \dots, x_n]$

For a point in the projective space $\mathbb{P}^n$, $p = [a_0:a_1:\dots :a_n]$, how to see that $$I(p) = \left < x_ia_j - x_ja_i: 0 \leq i \leq n, 0 \leq i \leq n \right> \subseteq K[x_0, ...
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For ideal $m$ maximal and principal, there's no ideal between $m^2$ and $m$. Prove that this can be false when $m$ is not principal or maximal.

Prove that for ideal $m$ maximal and principal, there's no ideal $I$ such that $m^2 \subsetneq I \subsetneq m$. Show that this can be false when $m$ is not principal or maximal. Suppose ...
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Find all maximal ideals of the ring $\mathbb Z_4 \oplus \mathbb Z_{15}$

Find all the maximal ideals of the ring $\mathbb Z_4 \oplus \mathbb Z_{15}$. The maximal ideal should be of the form $<1> \oplus <p>$ or $<p> \oplus <1>$ where $p$ is a ...
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The intersection and sum of irrelevant ideals are also irrelevant

Definition: A homogenous ideal $I \subset K[x_0,\dots,x_n]$ is irrelevant if $\left <x_0^r,\dots,x_n^r \right> \subset I$ for some $r > 0$. For $I \cap J$, this is probably circular logic, ...
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Describing integral ideals [closed]

Suppose I have a field $K=\mathbb{Q}(\sqrt{-d})$. How does one describe it's integral ideals?
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Prove that $J \cap R(1 - e) \neq \{0\}$

Let $R$ be a commutative unitary ring and I and J be two maximal ideals of $R$ such that $I \neq J$ and $I \cap J = Re$ where $e \in R$ is idempotent. Prove that $J \cap R(1 - e) \neq \{0\}$. I ...
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Verification of a factorization of ideals, $\langle 2 \rangle$

Still going over Alaca & Williams (I might die before I fully understand that book). In $\mathbb{Z}[\sqrt{-21}]$, the factorization of $\langle 2 \rangle$ is $\langle 2, 1 + \sqrt{-21} ...
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Chains of ideals contained in maximal ideal in non-Noetherian commutative ring

Given a maximal ideal $M$ in a non-Noetherian commutative ring $R$, I'm trying to determine whether or not there can exist infinite strictly ascending chains of ideals of $R$ contained in $M$. I know ...
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How to prove A is a flat R-module [duplicate]

Let A be a right R-module. Suppose for every left ideal J of R, the homomorphism $f:A\otimes J\to A$ defined by $f(x\otimes y)=xy$ is injective, then A is a flat R-module.(the identity 1 is in R) I ...
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Understanding properties of quotient rings

We have begun learning quotient rings in my Algebra course, but I am still confused by some of the theorems and properties of quotient rings. We have a theorem: $R$ is a ring, and $I\subset R$ ...
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Ideal of the union of two skew lines in $\mathbb{P}^{3}$

Let $$ L_{1}=V(X_{0},X_{1})\subseteq\mathbb{P}^{3}, $$ $$ L_{2}=V(X_{2},X_{3})\subseteq\mathbb{P}^{3}. $$ I want to prove that $$ I(L_{1}\cup L_{2})=(X_{0}X_{2},X_{0}X_{3},X_{1}X_{2},X_{1}X_{3}). $$ ...
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Is it true that if some power of an ideal is primary, then the ideal itself is also primary?

Is it true that if some power of an ideal $I$ is primary, then $I$ itself is also a primary ideal? I do not know whether the above statement is true or there is a counterexample. If one wants to ...
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What's so special about a prime ideal?

An ideal is defined something like follows: Let $R$ be a ring, and $J$ an ideal in $R$. For all $a\in R$ and $b\in J$, $ab\in J$ and $ba\in J$. Now, $J$ would be considered a prime ideal if ...
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Contraction and extension of ideals respect inclusions, sums and intersections

Let $R$ be an integral domain. Let $Y$ be a multiplicatively closed subset of $R$ which contains $1$ but not $0$. Define $S=RY^{-1}=\lbrace ry^{-1} : r \in R, y \in Y \rbrace$ as well as ...
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prove that maximal ideal in Z generated by a prime number

I am trying to prove <a> is a maximal ideal in Z, iff a is prime number. Now I wrote: assume a ∈ Z, while it's not prime number we can write as a=xy, for some integer x and y. then <a>⊂<x> ...
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Is $F[x]/\langle x^3-1\rangle$ a field?

Let F be a field. Is $F[x]/\langle x^3-1\rangle$ a field? I know $x^3-1 = (x-1)(x^2+x+1)$. How can I use this to show the above is a field or not? I am using the above to deduce $\langle ...
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Intersection of n hyperplanes in projective space of dimension n is not empty

I want to prove the following: Let $H_1,\dots,H_n$ be $n$ hyperplanes in $\mathbb{P}^n =\mathbb{P}^n \mathbb{C}$. Then $\cap_{i=1}^n H_i$ is not empty. Please be noted that this is an exercise ...
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Showing that an element generates the kernel

$I $ is a monomial ideal generated by $\left < m_1, \dots, m_n\right >$ and suppose we also have an $R$-module homomorphism $\phi: \oplus_{j = 1}^n Re_j \to I$ defined by $$\phi(e_i) = m_i.$$ ...
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Intersection multiplicity for two surfaces definded by $f=0,g=0$

I want to understand how I can find the intersection multiplicity $I_p$ at a point $p$ for two curves $f,g$. I have the example where $$ f(x,y) = y^2-x^3, \,\,\,\, g(x,y)=y^2-x^2(x+1) $$ Then I am ...
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$G(B) = \{k \in \Bbb{Z} : \#(A+k) \cap B = \infty \text{ whenever } \#(A \cap B) = \infty\, A \subset \Bbb{Z}\}$ is a group. Question…

Let $B \subset \Bbb{Z}$ be any infinite subset. Define $G(B) = \{k \in \Bbb{Z} : \#(A+k) \cap B = \infty \text{ whenever } \#(A \cap B) = \infty\, A \subset \Bbb{Z}\}$. It forms an additive subgroup ...
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Show that the prime ideals above a prime $p$ are principal

Let $K=\mathbb Q(\alpha)$ where $ \alpha^3 -5\alpha + 5 = 0 $. I need to show that the prime ideals above 5 are principal, and find a generator for them. I have worked out the prime decomposition of ...
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$R$ is isomorphic to product of ideals $(a)\times (1-a)$

Following a question of mine on here, the answer says that if $a\in R$ (a commutative ring) is idempotent ($a^2=a$), then: $$R\simeq (a)\times (1-a)$$ I am trying to make sense of this by proving it. ...
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Finding all ideals N of $Z_{12}$ and compute $Z_{12}/N$ in each case.

I already find all the ideals N of $Z_{12}$. Here's what I've got: Since ideals must be additive subgroups, by group theory we see that the possibilities are restricted to the cyclic additive ...
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Cardinality of the quotient of $M_2(\mathbb{Z})$ by the ideal of matrices with even entries

Let $R=\left\{\begin{pmatrix} a_1&a_2\\a_3&a_4 \end{pmatrix} \mid a_i\in \mathbb{Z} \right\}$ and let $I$ be the subset of $R$ with even entries. Show that $I$ is an ideal of $R$. What is ...
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Finding ideals of $F_2[C_2]$

I'm trying to find the ideals of $F_2[C_2]$ I believe the elements are $(0,1,x) $ So far I have the ideal {0} I can't seem to spot any others, have I made a mistake or missed something?
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Are there any uses of this notion of ideal? $xy + k \in (p) \implies x + k$ or $y + k \in (p)$

Abstracting the definition of prime ideal a little we have ideals $(p) \subset \Bbb{Z}$ such that $xy + k \in (p) \implies x + k \in (p)$ or $y + k \in (p)$. For example, taking $k = 1$, and $p = 2$ ...
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Show $(3,x)$ is a principal ideal in $\mathbb{Z}_{6}[x]$.

Since $R[x]/(I)=(R/I)[x]$, and by the Chinese Remainder Theorem, $$\mathbb{Z}_6[x]/(3,x) \cong \mathbb{Z}_3[x]/(3,x) \times \mathbb{Z}_2[x]/(3,x) \cong ...
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intersection of principal ideals in commutative rings with unity [duplicate]

Consider a commutative ring with unity. The intersection of two principal ideals is an ideal but not necessarily a principal ideal. (http://commalg.subwiki.org/wiki/Principal_ideal) However in ...
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Proving that particular ideal is the kernel of a homomorphism of polynomial rings

I have the homomorphism $f:\mathbb{R}[x,y,z]\rightarrow\mathbb{R}[t]$ with $f(x)=t,f(y)=t^2,f(z)=1$, and I want to prove that $\ker f=(x^2-y,z-1)$. It is obvious that this ideal is in kernel, but ...
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Compute the decomposition of $5\mathbb{Z}_K$ as a product of prime ideals

Let $K = \mathbb{Q}(\alpha)$ such that $\alpha^3 - 5\alpha + 5 = 0$. It is easy to show that $\mathbb{Z}_K = \mathbb{Z}[\alpha]$ and that $5$ is not maximal in $\mathbb{Z}[\alpha]$. So we cannot use ...
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$R$ be a commutative ring with unity such that every prime ideal contains no non-zero zero divisor , then is $R$ an integral domain?

Let $R$ be a commutative ring with unity such that every prime ideal contains no non-zero zero divisor (i.e. if $P$ is a prime ideal and $x,y \in P$ with $xy=0$ then either $x=0$ , or $y=0$). Then ...
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Compact operators form the only closed proper ideal of bounded linear operators

I am trying to understand the following proof in Trace Ideals and Their Applications by Barry Simon (Proposition 2.1): Let $\mathcal{J}$ be a two-sided ideal in $\mathcal{L}(\mathcal{H})$ containing ...
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Ideal $I=({ X }^{ 2 },2X)$ of $\mathbb Z[X]$ generated by ${ X }^{ 2}$ and $2X$ is not primary.

Let $I=({ X }^{ 2 },2X)$ ideal of $\mathbb Z[X]$ generated by ${ X }^{ 2}$ and $2X$. Show that $I$ is not primary. I tried to find $$\sqrt { I } =\sqrt { ({ X }^{ 2 })+(2X) } =\sqrt { \sqrt { { ...
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Proving that there is no ideal $I$ in $\mathbb{Z}[t]$ such that $\mathbb{Z}[t]/I \cong \mathbb{Q}$ [duplicate]

I want to prove that there is no ideal $I$ in $\mathbb{Z}[t]$ such that $\mathbb{Z}[t]/I \cong \mathbb{Q}.$ The first part of the question asks us to show that if $\phi$ is a nonzero ring ...
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Identities of the spectrum of a ring

Let $R$ be a commutative ring, then the spectrum of $R$ is a topological space: $$Spec(R)=\{ P:P \;\text{is a prime ideal of R} \}$$ where the close sets are $$V_{I}=\{P \in Spec(R) \;\text{such ...
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Example(s) of tertiary ideals?

Could anyone provide me with some specific example of a tertiary ideal, illustrating why it is so? Thanks in advance.
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Is it possible to know the prime factors of $3+5i$?

Let $\mathbb{Z}[i] = \{a+bi : a,b \in \mathbb{Z}\}$ and $\mathbb{Q}(i) = \{a+bi : a,b \in \mathbb{Q}\}$. Find $\alpha \in \mathbb{Z}[i]$ such that $(3+5i,1+3i) = (\alpha)$ Since ...