# 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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### Nilpotent or non-Nilpotent Jacobson Radical

Let $R$ be a ring with identity element such that every ideal of which is idempotent or nilpotent. Is it true that the Jacobson radical $J(R)$ of $R$ is nilpotent? If $R$ is Noetherian and $J(R)$ is ...
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### $R$ be a Noetherian domain , $t\in R$ be a non-zero , non-unit element , then is it true that $\cap_{n \ge 1} t^nR=\{0\}$?

Let $R$ be a Noetherian domain, $t\in R$ be a non-zero, non-unit element, then is it true that $$\bigcap_{n \ge 1} t^nR=\{0\} \text{?}$$ It almost feels like the nilradical (which is zero for any ...
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### $(R,\mathcal m)$ be a Noetherian local ring and let $P$ be a prime ideal of $R$. If $P^2$ is a prime ideal of $R$, then $P=0$

Let $(R,\mathcal m)$ be a Noetherian local ring and let $P$ be a prime ideal of $R$. If $P^2$ is a prime ideal of $R$, then $P=0$. I was thinking to use Nakayama lemma as: $R_P$ is local with $PR_P$ ...
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### How is a sequence not converging usually but $I_{\tau}$ converging in this given paper.

I am reading the paper Pratulananda Das and Ekrem Savas: On I-convergence of nets in locally solid Riesz spaces, Filomat 27:1 (2013), 89–94, DOI: 10.2298/FIL1301089D. I am stuck at example $3.2$ ...
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### Minimal ideal in a ring which is generated by an idempotent element.

Let $R$ be a commutative ring with unity and $M$ be a minimal ideal of $R$ such that $M = Re$ where $e$ is an idempotent element in $R$. Then $R = Re \oplus R(1-e)$ I am not able to see, in order ...
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### $R$ be an integral domain , $x \in R$ , $I$ an ideal such that $I+\langle x \rangle , (I:x)$ are principal ideals , then is $I$ a principal ideal?

Let $R$ be an integral domain , $x \in R$ , $I$ be an ideal such that $I+\langle x \rangle$ and $(I:x):=\{r \in R : rx \in I\}$ both are principal ideals , then is $I$ also a principal ideal ?
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### What are the conditions needed for two principal ideals of a ring to be isomorphic?

Given a commutative ring $R$, and $p(x),q(x) \in R[x]$ monic polynomials, under what conditions on $p(x)$ and $q(x)$ are the principal ideals $\langle p(x) \rangle$ and $\langle q(x) \rangle$ ...
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### Polynomial in Compact Polytope: Algebraic Description for the Compact Polytope?

Consider a polynomial $f\in K[x_1,\ldots, x_n]$ where $K=\mathbb R$. For example $$f[x_1,x_2,x_3]=x_1 x_2+x_3$$ \begin{eqnarray*} x_{1} & \in & [0.2,0.5]\\ x_{2} & \in & [0,1]\\ x_{3}...
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### Show that the ring of integers $A$ of the cubic field $\mathbb Q[x]$ with $x^3=2$ is principal.

Show that the ring of integers $A$ of the cubic field $K=\mathbb Q[x]$ with $x^3=2$ is principal. The hint given in the book is to majorize the discriminant of $A$ by $D(1,x,x^2)$ and then use the ...
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### Obtain dimension of multivariate polynomial quotient ring?

Let $\mathbb{C}[z_1,z_2,...,z_n]$ be the ring of multivariate polynomials in complex variables $z_1,z_2,...,z_n$ with complex coefficients. This ring is spanned by the countably infinite basis of ...
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### Uniqueness of generator of principal ideal in K[x_1,x_2,…,x_n]

In $K[x]$ (where $K$ is a field), I know that every ideal can be written as $(f)$ for some $f \in K[x]$. Furthermore, $f$ is unique up to multiplication by a nonzero constant in $K$. Is there a ...
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