# 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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### Prove $(y-x^2)$ is a prime ideal in $\mathbb{R}[x,y]$, but not maximal.

My guess is to use the fact that when we take the quotient, $\mathbb{R}[x,y]/(y-x^2)$, this will become an integral domain but not a field. I am not sure how to take the quotient, though. I am also ...
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### example of right / left ideals

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 ...
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### When do two ideals have trivial intersection?

Let $R$ be a ring and $I,J$ two ideals. If $I \cap J=0$ then $ij=0$ for every $i \in I$ and $j \in J$. This happens when $R=A \times B$ and $I=I’ \times \{0\}$ and $J=\{0\} \times J’$ with $I’$ ideal ...
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### Radical of the ideal generated by $x^3 - y^6$ and $xy-y^3$

I am following Andreas Gathmann's notes on algebraic geometry. He asks, right after he shows that $V$ and $I$ are bijection between algebraic varieties and radical ideals (so I suppose I must use ...
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### Image of an ideal under a surjective ring homomorphism is an ideal

Let $\phi: R \longrightarrow R'$ be a surjective ring homomorphism and $I$ an ideal in $R$. Show that $\phi(I) = \{ \phi (r) : r \in I \}$ is an ideal in $R'$. So I asked this question a couple ...
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### Prove that the ideal of $\mathbb{R}[x]$ generated by $x^3-x^2+x-1$ and $x^4+3x^2+2$ is a prime ideal [duplicate]

Prove that the ideal of $\mathbb{R}[x]$ generated by $f(x)=x^3-x^2+x-1$ and $g(x)=x^4+3x^2+2$ is a prime ideal. Also, prove that the ideal generated by $r(x)=x^3-x^2-x+1$ and $s(x)=x^4+x^2-2$ Call ...
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### Polynomial rings and ideals

I'm trying to learn some algebra by myself and I need some help. $I=(x^2,x^3)$ an ideal in $R[X]$. Give an example of two polynomials with exactly four terms, one that is in $I$ and one that isn't. ...
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### Showing a set is an ideal in a ring of real-valued functions

If $F$ is a ring of all real-valued functions defined on $\mathbb{R}$, is $S = \{f ∈ F | f(0) = 1\}$ an ideal? What I'm thinking is $(f+g)(0) = f(0)+g(0) = 1+1 = 2$ and hence $f + g$ is in $S$? Is ...
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### Describing elements in a principal ideal in a ring that does not have a $1$.

Let $R$ be any ring that does not contain $1$. For $a\in R$, describe the elements of $(a)$. Breaking this down piece by piece. Since the ring $R$ does not contain $1$, then the ring does not contain ...
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### Finding elements of the principal ideal $(10)$.

Let $R=2\mathbb{Z}$ be a commutative ring that does not contain $1$. What elements of $(10)$ are not in $10R$? $R=2\mathbb{Z}=\{0,\pm 2,\pm4,\dots\}$ So, $10R=\{0,\pm 20,\pm40,\dots\}$ So am I ...
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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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### If $A$ and $B$ are ideals of a ring $R$. Then $A+B$ is an ideal of $R$ generated by $A \cup B$? [closed]

I have proved that $A+B$ is an ideal of $R$. But I'm not able to prove that it is generated by $A \cup B$.
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### Help with proof of $\mathbb{C}[X] \simeq R$ where $R$ is a $\mathbb{C}$-algebra without nilpotents

I am trying to understand the proof of the following proposition: Let $X \subset \mathbb{A}^n$ be closed. Let $R$ be a finitely generated $\mathbb{C}$-algebra without nilpotents. There exists an ...
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### Does $S^{-1}I \subset S^{-1}J$ imply $I \subset J$?

Let $S$ be a multiplicative subset of a commutative ring with identity, and consider the ring of fractions $S^{-1}R$. Ideals in $S^{-1}R$ of are of the form $S^{-1}I$, where $I$ is an ideal in $R$. ...
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### By showing that $R/I$ is an integral domain, deduce that it's a field.

Let $R = \mathbb{F}_3[x]$ and let $I$ be the set $\{ (x^2+1)p(x)|p(x) \in R \}$ which is an ideal of R. By showing that $R/I$ is an integral domain, deduce that $R/I$ is a field. I know that $R/I$ is ...
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### Definition of principal ideal in rings [closed]

Can an improper ideal ($\varnothing$ or $R$) be a principal one in the ring $R$?
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### How to prove that a ring ideal is not principal?

I need to prove that an ideal $(x+1, y)$ in the ring $\mathbb{Q}[x,y]$ is not principal. I already tried to prove the statement by contradiction supposing that $(x+1,\ y)$ is principal. So, here is ...
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### is the ideal $(x-y,x+y)$ same as $(x,y)$

Is the ideal $(x-y,x+y)$ same as $(x,y)$, since $$x+y, x-y \in \mathbb{C},$$ so $y \in \mathbb{C}$ because $\mathbb{C}$ is a field. And similarly $x$ in $\mathbb{C}$, so $(x,y)\in (x-y,x+y)$ ...
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### Closed ideal in $L^{1}(G)$

Let $G$ be locally compact group prove that $$L_{0}^{1}(G)=\left\{f\in L^{1}(G): \int_G f(g) dm(g)=0 \right\}$$ is a closed ideal in $L^{1}(G)$ with codimension one I am grateful for any ...
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### Determining ideal of a ring [closed]

In general, can someone please explain how to determine what the ideals of a ring are? I understand that an Ideal is a subset of a ring such that it contains any element in the ring multiplied by the ...
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### Isomorphism between quotient rings and modules

Let $I_1, I_2$ be ideals of $R$ — associative ring with unit. Find an example where $R/ I_1$ and $R/I_2$ isomorphic as rings, but not isomorphic as modules. Can you check my solution? I have an ...
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### Smallest Ideal in $M_2(Z)$

What is the smallest ideal in $M_2(Z)$ containing $\begin{pmatrix} 1 & 0 \\ 0 & 0\end{pmatrix}$? I'm a bit unsure about what "smallest" means here. I've found all of the ideals of $M_2(Z)$, ...
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### Is $\{p(x) ∈ \Bbb Q[x]\mid p(0) = 3\}$ an ideal of $\Bbb Q[x]$? [closed]

Is $\{p(x) ∈ \Bbb Q[x] \mid p(0) = 3\}$ an ideal of $\Bbb Q[x]$? I don't have any idea of how to start this problem. Any help would be great, thank you in advance!
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### About comment by Jacobson on proving that a morphism in $\mathbf{Ring}$ is monic iff it is injective

I am reading Nathan Jacobson's Basic Algebra II, Chapter 1 Categories, and in $\S$2 Some Basic Categorical Concepts, he introduces the notion of a morphism being monic or epic. He asks the reader to ...
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### In a Euclidean ring $R$, prove $(a) ⊆ (b) \iff b|a$

Let $a, b$ be elements of a Euclidean ring $R$. Prove that $$(a) \subseteq (b) \iff b \;\text{divides}\;a.$$ I have no clue how to even start this. Any help would be great, thank you in advance!