# Questions tagged [ideals]

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, ideals in Lie algebras.

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### Constructing rings with a specific lattice of ideals.

Let $R$ be a commutative ring with 1. The ideals of $R$ form a lattice with inclusion as order relation. Let me call it the ideal lattice $L(R)$ of $R$. Given an arbitrary lattice $L$, there are some ...
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### Short Question: $(p)$ for a prime is not a maximal ideal in $\mathbb{Z}[X]$

Given a prime number $p\in\mathbb{Z}$ I want to show that $(p)$ is not a maximal ideal in the ring of polynomials $\mathbb{Z}[X]$. I know how the maximal ideals in $\mathbb{Z}[X]$ look like, I want ...
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### On the maximal ideals of $\Bbb Z_5[X,Y]$ which contain $\langle Y \rangle$

Let $R:=\Bbb Z_5[X,Y]$ and $I:=\langle Y \rangle \trianglelefteq R.$ 1) Prove that $I$ is prime but not maximal ideal. 2) Find all maximal ideals of $R$, which contain $I$. Answer. 1) If we take ...
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### Converse of : If $M$ is Noetherian then $M_m$ is Noetherian for every $m\in\operatorname{Max}(R)$.

Let $R$ be a Noetherian ring and $M$ be an $R$-module. We know that if $M$ is Noetherian then $M_m$ is a Noetherian $R_m$-module for every $m\in\operatorname{Max}(R)$. Now, I want to know if the ...
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### Show that if $I$ is an ideal contained in $J$ then $(R/I)\setminus (R/I)^{\times}$ is an ideal of $R/I$

Question: Let $R\neq 0$ be a ring and $J= R\setminus R^{\times}$ an ideal of $R$. Show that if $I\subset J$ is an ideal of $R$ then $(R/I)\setminus (R/I)^{\times}$ is an ideal of $R/I$ Attempt: I ...
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### The ideal generated by a maximal orthogonal system in a Banach lattice

I have a pretty specific question about H.H. Schaefer's "Banach lattices and positive operators" book. In chapter 3, part 6 (page 169), it is said that the ideal generated by a set S (which is a ...
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### Ideals of $R[x]/I[x]$, where $I$ is a maximal ideal of $R$

Original Question: Let $R$ be a commutative ring with identity and $I$ maximal ideal in $R$. Show that $I[x]$ is a prime ideal in $R[x]$ and is not maximal ideal in $R[x]$, find two distinct maximal ...
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### Is it still unknown whetever any $\mathscr{D}$-class $D$ being a semigroup is bisimple?

Introduction: It is stated in my book from year 1961 that it is unknown whatever a subsemigroup $D$ of semigroup $S$, which is also a $\mathscr{D}$-class of $S$ is always bisimple, but it is known ...
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### Contraction of quotient ideal is quotient of contractions?

Let $\mathfrak a,\mathfrak b$ be ideals in a ring $A.$ The quotient of $\mathfrak a$ and $\mathfrak b$ is $(\mathfrak a:\mathfrak b)=\{x\in A:x\mathfrak b\subseteq \mathfrak a\}$ and if $f:B\to A$ is ...
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### Finding a prime ideal whose class is order 10 in the ideal class group

"By factorising the ideal $(4+\sqrt{-74})_{R}$, or otherwise, find a prime ideal whose class $[P]$ in the ideal class group $Cl(R)$ has order 10" $R = \mathbb{Z}[\sqrt{-74}]$ So I've managed to get ...
Let $K \subset L$ be a field extension, $K[X]$ and $L[X]$ the corresponding polynomial rings (in one variable) and $I \subset K[X]$ an ideal. I want to show that $I=K[X] \cap IL[X]$, where $IL[X]$ ...
I searched and searched for examples of right / left ideals, but could find none. I read that a right ideal of $S$ is a subset of $R$ of $S$ such that $RS \subseteq R$, and that symetrically, a left ...