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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Ideals generated by roots of polynomials

Let $\alpha$ be a root of $x^3-2x+6$. Let $K=\mathbb{Q}[\alpha]$ and let denote by $\mathscr{O}_K$ the number ring of $K$. Now consider the ideal generated by $(4,\alpha^2,2\alpha,\alpha -3)$ in ...
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Ideal of the Integers

Show that an ideal $(m)$ of $\mathbb{Z}$ is maximal if and only if $m$ is prime. I know that if $m$ is a prime number. And $\mathbb{Z}/m$ is a field and thus $m$ is maximal. I know i have to do a ...
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65 views

Number of ideals in $\Bbb Z[x]/(x^3+1, 7)$

I am trying to find the number of ideals in $R:=\Bbb Z[x]/(x^3+1, 7)$ and $S:=\Bbb Z[x]/(x^3+1, 3)$. I started with $R$ and tried to write it in terms of familiar rings, by using fundamental ...
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$(\Bbb Z[x]/(x^{n+1}))^\times\cong\Bbb Z/2\Bbb Z \times \prod _{i=1}^n \Bbb Z$ [duplicate]

I'm trying to prove the group isomorphism $(\Bbb Z[x]/(x^{n+1}))^\times\cong\Bbb Z/2\Bbb Z \times \prod _{i=1}^n \Bbb Z$. Obviously I tried to establish a ring isomorphism from $\Bbb Z[x]/(x^{n+1})$ ...
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Show the ideal $I=(x^{2}-y,z-1)$ is prime in $K[x,y,z]$

I am tasked to show that the ideal $I=(x^{2}-y,z-1)\subset K[x,y,z]$ is it's own radical where $K$ is an algebraically closed field. I tried to proceed in the obvious fashion. Let ...
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Finding the kernel of ring homomorphisms from rings of multivariate polynomials

I am trying to find the kernels of the following ring homomorphisms: $$ f:\Bbb C[x,y]\rightarrow\Bbb C[t];\ f(a)=a\ (a\in\Bbb C),f(x)=t^2,f(y)=t^5. $$ $$ g:\Bbb C[x,y,z]\rightarrow\Bbb C[t,s];\ g(a) = ...
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Integral Ideals

Any nonempty set of integers $J$ that fulfills the following conditions is called an integral ideal:   (i)  If $n$ and $m$ are in $J$, then $n+m$ and $n-m$ are in $J$. ...
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Showing Quotient Ring is a Field

Consider the ring $S=\mathbb{Z}[\alpha]$, where $\alpha = \sqrt[3]{2}$, and ideal $I=(5,\alpha^{2}+3\alpha -1)$. I wish to show that $S/I$ is a field of order 25. Any solutions/suggestions? I would ...
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Some property of an ideal in commutative ring [duplicate]

Let $R$ be a Dedekind domain, and let $\mathfrak{m}$ be a maximal ideal in $R[x]$ is of the form $\mathfrak{m} = (\mathfrak{p},f(x))$ where $\mathfrak{p}$ is a maximal ideal in $R$, and $f$ is a ...
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A (probably) wrong exercise from Morandi's Field and Galois theory

After some efforts I realize that the following exercise is wrong: (rings are unitary throughout the book) Morandi's Field and Galois Theory, Appendix A, exercise 18 (b) Let $A\subseteq B$ ...
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Left and Right Ideal Generated by Two Matrices.

Let $R= {\rm Mat}_2(\Bbb R)$ be the ring (with $1$) of $2\times2$-matrices with entries in $\Bbb R$. Let $$M = \left\{\begin{pmatrix}1&0 ...
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About quotient ring

I want to find the value $|\mathbb{Z}({\sqrt{2})/(3+\sqrt{2})}|,|\mathbb{Z}({\sqrt{13})/(5+\sqrt{13})}|$ and also the number of ideals of $\mathbb{Z}({\sqrt{13})/(5+\sqrt{13})}$. But still not ...
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Is zero in every ideal?

Let $I \subseteq R$ be a subset of a ring $(R, +, \cdot)$ such that $(I, + )$ is a group $\forall x \in I, r \in R: xr \in I \text{ and } rx \in I$ Then $I$ is called an "Ideal" Question: Is 0 in ...
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$\langle x^a, y^b\rangle $ is an irreducible ideal in $K[x,y]$

Prove that $\langle x^a, y^b\rangle$ is an irreducible ideal in $K[x,y]$. Any kind of help is very much welcomed.
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Vandermonde matrices over a commutative ring.

Suppose that $R$ is a commutative ring with identity. I am trying to prove that the two following statements are equivalent. The ideal generated by all determinants of $n\times n$ Vandermonde ...
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Homomorphism between two algebras

So I know from ring theory that if we have a homomorphism $\phi: A \rightarrow B$ then it must be the case that for some ideal $I \subseteq A$ we have $\phi(I) \subseteq B$ and $\phi(I)$ is an ideal ...
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Maximal ideal in commutative ring

Let $R$ be a Dedekind domain, $\mathfrak{m}$ be a maximal ideal in $R[x]$ is of the form $\mathfrak{m} = (\mathfrak{p},f(x))$ where $\mathfrak{p}$ is a maximal ideal in $R$, and $f$ is a polynomial in ...
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234 views

Prove that the ideal $(X+Y+1)$ is prime in $F[X,Y]$

Let $F$ be a field. Prove that the ideal $I=(X+Y+1)$ is a prime ideal in the polynomial ring $F[X,Y]$.
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Isomorphism of quotients of powers of maximal ideals

Let $R$ be an integral domain, and $\mathfrak{m}$ a maximal ideal of $R$. Let $R_\mathfrak{m}$ denote the ring localized at $\mathfrak{m}$, and let $\mathfrak{m}_\mathfrak{m} = ...
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Expressing ideals as products of prime ideals in a commutative, Noetherian ring with unity

Let $R$ be a commutative, Noetherian ring with unity. I know that the following is true: For any ideal $I\subset R$, there are prime ideals $\mathfrak{p}_1,\ldots,\mathfrak{p}_n$ such that ...
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Counting the ideals of $\frac{\mathbb{R}[X]}{(X^2)}$

I want to ask you guys if I'm on the right track: Here's the question: Suppose $a \in \mathbb{R}$. Count the ideals of $\frac{\mathbb{R}[X]}{(X^2-a)}$. Give an example of a ring with exactly 3 prime ...
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Let R be a commutative ring with identity. Prove: every nilpotent is inside every prime ideal. [duplicate]

Let R be a commutative ring with identity. Prove: every nilpotent is inside every prime ideal. Don't know what to use with this problem. Any help is usefull :)
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Ideals in the polynomial ring over a division ring are free

Let $k$ be a division ring. I want to show that every (right) ideal in $k[x]$ is free considered as a right $k[x]$-module. That means if $I$ is an ideal in $k[x]$ we have to show that $I=f\cdot ...
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$R/I$ when $R$ is the ring of real continuous functions

If $R$ is the ring of all real continuous functions on $[0,1]$, I am trying to find $R/I$ where $$I=\{f\in{R}|f(.5)=0\}$$ Showing $I$ is an ideal is not a problem since we're defining addition and ...
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202 views

Prove that in any integral domain there is at least one prime ideal.

How can I prove that every integral domain has at least one prime ideal? I don't know if I'm overthinking it, but maybe I am. I know how to prove it for something like $\Bbb Z$, but I'm not sure how ...
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Ideal of a ring

I'm trying to describe an ideal of the ring $R=\left\{ \begin{pmatrix}a & b\\ 0 & c \end{pmatrix}:a,b,c \in \mathbb{R}\right\} $ It's easy to prove that $I=\left\{ \begin{pmatrix}0 & a\\ 0 ...
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Primes for which $\Bbb Z[i]/p\Bbb Z[i]$ is a field [closed]

Determine (with proof) all prime numbers $p$ for which the ring $\Bbb Z[i]/p\Bbb Z[i]$ is a field.
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478 views

Verification of a proof that the ring of continuous functions on $[0,1]$ is not noetherian

My problem is to show that the ring of continuous real-valued functions on the interval $[0, 1]$ is non-noetherian. I'd like to know if this argument holds: Let $I$ be the set of functions such ...
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Chain conditions satisfied by some algebraic structures including ideals in certain commutative rings

From http://en.wikipedia.org/wiki/Ascending_chain_condition: "In mathematics, the ascending chain condition (ACC) and descending chain condition (DCC) are finiteness properties satisfied by some ...
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Closed ideal in the hereditary C*-subalgebra

Let B is a hereditary C*-subalgebra of a C*-algebra A and J be a closed ideal of B, is AJA a closed ideal of A? I do not find the definition of product of ideals in the C*-algebras, is it the same ...
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A prime ideal $I$ in $\mathbb Z[\sqrt{-5}]$ so that $7\in I$.

Find a prime ideal $I$ in $\mathbb Z[\sqrt{-5}]$ such that $7\in I$. I claimed that $I= 7\mathbb Z[\sqrt{-5}]$, and tried to prove that if $x,y\in \mathbb Z[\sqrt{-5}]$ and so that $xy\in I$ then ...
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Cardinality of the quotient ring $\mathbb{Z}[x]/(x^2-3,2x+4)$

This problem is from a practice exam I was working on. What is the cardinality of the quotient $\mathbb{Z}[x]/(x^2-3,2x+4)$ ? Thoughts. If I find a ring that is easier to handle then this then I ...
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1answer
91 views

Three quotient-ring isomorphism questions

I need some help with the following isomorphisms. Let $R$ be a commutative ring with ideals $I,J$ such that $I \cap J = \{ 0\}$. Then $I+J \cong I \times J$ $(I+J)/J \cong I$ $(R/I)/\bar{J} ...
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Some homework about ideals

I will ask you some things about ideals. Determine weither the ideal $(X^2+3) \subset \mathbb{F}_5[X]$ is maximal or prime. Intuitively I'd say that the ideal is prime but not maximal. To prove ...
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If $I\subseteq J\subseteq A$ have same image in localization by all maximal ideals, then $I=J$

I will state my question first: Suppose $I\subseteq J\subseteq A$ are two ideals in a commutative ring $A$. Furthermore, assume that for every maximal ideal $\mathfrak{m}$ of $A$, the image of ...
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How to show that $\sqrt{\{0\}} = \bigcap_{\text{I is prime}}I$ in a commutative ring with unit

Here's another question about ring theory. How to show that $\sqrt{\{0\}} = \bigcap_{\text{I is prime}}I$ in a commutative ring with a unit. Own attempts For "$\subseteq$", let $x$ be an arbitrary ...
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1answer
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Semigroup which satisfied $(xy)^{\pi} = (xy)^{\pi}x$ and principal right ideals

Let $(S,\cdot)$ be a finite semigroup, then every $s \in S$ has an unique idempotent power, i.e. there exists a smallest $i \in \mathbb N$ such that $s^i$ is idempotent. It is the unique idempotent in ...
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Codimension of ideals in polynomial rings over PIDs

Let $R$ be a (commutative) principal ideal domain and let $J$ be an ideal in $R[x_1,\dots,x_n]$. Is it possible to make a general statement about the codimension (=rank/height) of $J$? As $J$ does ...
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Where in the proof did Herstein use the fact that $A$ is a two-sided ideal of $R$?

I'm reading Noncommutative Rings by I. N. Herstein. The theorem I'm having trouble with is 1.2.5, on page 16 of the book. Some definition 1. Regular ideal An ideal $\rho \subset R$ is called a ...
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$I=\{f\in \mathbb{C}[x,y,z,t] : f(-2,-1,1,2)=0\}$. Generators for $I$?

This problem is from a past qualifying exam I am trying to work on. I am stuck on trying to find generators for $I$. The question is as follows Let $\mathbb{C}[x,y,z,t]$ be the polynomial ring ...
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Ideal generated by two elements that is not free

I am having problems understanding an example constructed by M. Ojanguren and R. Sridharan showing that over the polynomial ring in two variables over a division ring (which is not a field) there ...
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107 views

Sum of ideals in the polynomial ring

Could someone explain to me how to find a sum of ideals where $I=(x+y)$ and $J=(x)$? The answer to this is $I+J=(x,y)$ and we work in the polynomial ring $k[x,y]/(xy)$. I know that the definition ...
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Isomorphic factor rings of polynomial rings does imply isomorphic ideals?

Let $k$ be a field, $I$ and $J$ are ideals of $R=k[x_1,\dots,x_n]$. If $R/I\simeq R/J$ as rings, then $I \simeq J$ as $R$-modules holds? Thanks in advance!
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Non Classical Examples of Indecomposable Ideals

A classical example of a ring $R$ with an indecomposable ideal is the ring $C(X)$ of real valued continuous functions on $X$, where the $(0)$ ideal is not decomposable. Does anyone know other examples ...
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Reading multiplicity of cusps , singularity etc from initial polynomial.

Here I have an example which I found. Can someone help me to understand what's happening here? The following are my concerns: 1) What do we need to do co-ordinate transformation? 2) How does the ...
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An ideal in the ring of infinitely differentiable functions

Well, I was just doing an elementary exercise, but am a tad bit skeptical about how I've gone about it. It goes as follows: Let $R$ be the ring of infinitely differentiable functions defined on, say, ...
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Kronecker's approach to unique factorisation in algebraic number theory: books and references

I have done a short course (one semester) on algebraic number theory at beginning graduate level, in which the Dedekind theory of ideals features prominently. However I have since discovered that ...
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Ideals in ring extensions

Let $R$ be a ring, commutative with $1$, subring of a ring $R'$. Let $\mathfrak{p}$ be an ideal of $R$. Let's denote by $\mathfrak{p}R'$ the extended ideal, i.e. the ideal generated by $\mathfrak{p}$ ...
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A question on generators

Suppose $I$ is an ideal in a ring $R$ which is finitely generated. Suppose on the other hand that there is some (possibly other) set of generators $\{g_t\colon t\in T\}\subset I$ which also generates ...
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Prime ideals and epimorphism

Let $\phi$:$R$$\rightarrow$$S$ be a ring epimorphism. Show that if $P\triangleleft S$ is a prime ideal (of S), then $\phi^{-1}(P)\triangleleft R$ is a prime ideal (of R). Can someone help me with ...