# Tagged Questions

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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### Radical of the powers of an ideal

I am asked to prove the following: $$\sqrt{\mathfrak{a}^n} = \sqrt{\mathfrak{a}}$$ Here is my attempt so far: $\sqrt{\mathfrak{a}^n} \subseteq \sqrt{\mathfrak{a}}:$ (By Induction) Clearly the ...
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### Sum of Ideals of the Same Type

I have two questions: 1) Is a finite sum of idempotent ideals of a ring $R$ idempotent? 2) Is any sum of nil ideals of a ring $R$ nil? As far as I know, a finite sum of nil ideals of a commutative ...
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### Show that the Group Ring $F_p[G]$ where $G$ is a $p$-Group has a unique maximal ideal.

Show that the Group Ring $F_p[G]$ where $F_p$ is finite field of order $p$ and $G$ is a $p$-Group (not necessarily abelian) has a unique maximal ideal, i.e. it is a local ring. Attempt: Consider ...
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### Contraction of an ideal

Let $f: \mathbb{Z}[X] \longrightarrow \mathbb{Z}[\sqrt{2}]$ be a ring homomorphism sending $X$ to $\sqrt{2}$. I am asked to compute a few contractions, and I am wondering if I could get some help in ...
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### Existence of minimal prime ideal contained in given prime ideal and containing a given subset

Let $R$ be a unital commutative ring, $P$ $\subseteq$ $R$ a prime ideal, $X\subseteq P$ a subset. Show there exists a minimal (inclusion minimal) prime ideal contained in $P$ which contains $X$. My ...
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### Understanding the concept of polynomial ideals

I am not able to understand the fundamental concept behind a polynomial ideal. What I have so far in terms of $I$ being an ideal of a ring is: for each $f, g \in I$, we have $-f$ and $f+g \in I$...
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### Ideal proof in a ring R

Problem Statement: Let I be an ideal in a ring R. Prove that K is an ideal, where $K$ = { $a\in R$ | $(\forall r\in R)(ra\in I)$} What exactly am I supposed to show here? I know I need to show ...
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### Let $I$ and $J$ be ideals in $R$. Is the set $K= \{ ab \ | \ a\in I, b\in J \}$ an ideal in R? [duplicate]

I've just assumed that this is false, since the problem statements says to compare it to a previous problem where $\{ a+b \ | \ a\in I, b\in J \}$ is ideal. However, by trial and error I can't find ...
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### Radical ideals in $\mathbb Z[x]$ such that their sum is not radical

I am trying to solve an exercise in which I have to provide an example of two radical ideals $I,J \subset \mathbb Z[x]$ such that their sum $I+J$ is not radical. I don't know how to attack this ...
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### If $R$ is a domain and $M_n(R)$ is semisimple, then $R$ is a division ring.

From Lam's A First Course in Noncommutative Rings, section 1.3. Let $R$ be a domain (EA: that is, a ring without zero divisors) such that $M_n(R)$ is semisimple. Show that $R$ is a division ring. ...
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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" generators, ...
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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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### 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 ...
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 $F[x]$...