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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If $\phi:R\rightarrow R'$ is a surjective ring homomorphism and I is an ideal in R… continued below [duplicate]

then $\phi(I)=[s'\in R'|s=\phi(s)\space \forall\space s\in I]$ is an ideal in R' So I know that for something to be an ideal, it needs to be closed under subtraction and it must absorb products. I ...
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Calculate generators of an intersection of Ideals

$$I = (x_1^2-x_1,x_2^2-x_2,...,x_n^2-x_n,t-\sum_{i=1}^n 2^{i-1}*x_i)$$ Ideal in $\mathbb Q[x_1, ..., x_n,t]$. How can I calculate the generators of $J = I \cap \mathbb Q[t]$ by hand? I tried it with ...
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Prove: If $\mathit R$ is a commutative ring with unity and $\mathit I=(x)\subseteq R[x]$, then $R[x] / (x)\cong R$ [duplicate]

I know that to show a ring is isomorphic to another ring, I have to find a bijective ring homomorphism between the two rings. Or I could use the F.H.T. but I would also need a function to make that ...
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Transitivity of integral extensions and prime ideals

The situation is as follows: We have $K$ a field $K[a_1, ..., a_n] \subseteq R$ finite ring extension $R \subset R'$ integral ring extension of integral domains Since $R$ is finite over $K[a_1, .....
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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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How do I find the generator $(g)$ that generates $(29, \sqrt{-5} ± 13)$

Find the generator $(g)$ that generates $(29, \sqrt{-5} + 13)$. The ring is $\mathbb{Z}\left[\sqrt{-5}\right]$. The fact I used was that $\text{Norm}(g)$ must divide both 29 and $\text{Norm}(\sqrt{5} ...
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Show that the product of ideals is equal to the intersection

I am following the notes of Gathmann to learn myself about commutative algebra. I have the following exercise written in them (without solutions at the end): Exercise 1.13. Show that the equation ...
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Isomorphism of Ideal tensored with affine open and restriction of ideals

Let $f:X \rightarrow Y = \operatorname{Spec}A$ be a morphism and $Y = \bigcup U_\alpha$ where $U_\alpha = \operatorname{Spec}A_\alpha$. Given the ideal $I = \{a\in A: f^*(a) = 0\}$, show that $I \...
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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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Sum and Intersection of Contraction and Extension

For Rings $X, Y$ with a ring homomorphism $f: X \rightarrow Y$. Let $\mathfrak{a}_1, \mathfrak{a_2} \subseteq X, \mathfrak{b}_1, \;\mathfrak{b_2} \subseteq Y$ be ideals. I was able to prove: $$\...
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Associated graded ring of an ideal is reduced, then the ideal is normal.

Let $I$ be an ideal in a Noetherian ring $R$ such that $\mathrm{gr}_I(R)$ is a reduced ring. Prove that $I$ is normal. What is the error in this proof? We proceed by induction over $n$. If $n=1$ we ...
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A characterization of principal rings

I would like to know if the following characterization (for principal rings, not necessarily domains) is true: $ A $ is principal $ \leftrightarrow $ $ A_\mathscr{M} $ is principal $ \forall \mathscr{...
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Finding units of a quotient ring.

I've been given a two part question to first find the number of elements and then number of units is in $R/I$ where $R = \mathbb{F}_7[x]$, $f(x) = x^3 + 4$ and $I = f(x)$ I found the number of ...
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Maximal ideals in $\operatorname{End}_A(M)$

$\newcommand\End{\operatorname{End}}$Let $A=k[x_1, \dotsc, x_n]$ and $M$ be a graded $A$-module. Let $E=\End_*(M)$ be the graded endomorphism ring of $M$ (i.e., $E_i$ consists of all degree-$i$-...
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Prove that a maximal ideal is radical.

Let $m$ be a maximal ideal of commutative ring $R$. Prove that $m$ is radical. I understand that $m$ is maximal if it is proper and there are no other ideals (except $R$) that properly contain it. ...
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Prove : product of two ideals is also an ideal

I'm new to Lie algebras. Let $\mathfrak{g}$ be a Lie Algebra with the braket $[a,b]=ab-ba$. and $I$ and $J$ are two Ideals, which means that: $$[I,\mathfrak{g}]\subset I$$ and $$[J,\mathfrak{g}]\...
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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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C(0, 1) has infinitely many prime ideals not of the form Mp [duplicate]

I know $\mathcal C([0, 1])$ has all maximal ideals of the form $M_p=\{f\in \mathcal C([0, 1]) : f(p) =0,\ p\in [0, 1] \}$. My question is little bit different. If I replace $[0, 1]$ by $(0, 1)$ ...
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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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Equations for vanishing set of ideal in coordinate ring.

Let there be ring $A = \mathbb{C}[x_{ij}, y_{ij}| i,j \in {1,2}] $ It's coordinate ring of 8 variables. Consider $I$ generated by $4$ elements $x_{11}y_{11}+x_{12}y_{21} , x_{11}y_{12} + x_{12}y_{22}...
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Prove that every prime ideal that isn't maximal is a minimal prime ideal

Suppose that the additive group of the ring $R$ is a finitely generated abelian group. If $P$ is a maximal ideal of $R$, show that $R/P$ is a finite field. Show that every prime ideal of $R$ that is ...
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Let J be an ideal. Find a function in I(V(J)) such that the function f is not in J

Let $J$ be the ideal $\langle x^2+y^2-1,y-1\rangle$. Find $f \in \textbf{I}(\textbf{V}(J))$ such that $f \not \in J$. I'm confused on a number of aspects here. Firstly, how do I find $\mathbf{V}(J)$ ...
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Determining the ideals of a quotient ring

Given an ideal $I = \langle x^3 - x\rangle \subseteq \Bbb{R}[x]$, determine the ideals in the quotient ring $\Bbb{R}[x]/I$. I understand that the quotient ring is of the form $k[x_1...x_n]/I$ where ...
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Proving that if coprime $\alpha_{i}\in R$ divide b, then $\alpha_{1}…\alpha_{n}$ divide b.

Let $R$ be a principal ideal domain and let $\alpha_{1},...,\alpha_{n}\in R$ be such that $(\gcd(\alpha_{i},\alpha_{j}))=(1)$. Let $b\in R$ such that each $\alpha_{i}|b$. I want to show that $\...
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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!
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Proof of principle ideal

Let $I = \{p(x) ∈ \Bbb Z[x]: 5\mid p(0)\}$. Prove that $I$ is an ideal of $\Bbb Z[x]$ by finding a ring morphism from $\Bbb Z[x]$ to $\Bbb Z_5$ with kernel $I$. Prove that $I$ is not a principal ideal....
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Prove that $Z_{14}/(7)$ ≅ $Z_7$

Prove that $Z_{14}/(7)$ ≅ $Z_7$. I understand that I have to show that there exists a surjective function relating the two and use the morphism theorem, but I'm not sure how in this case. Any help ...
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Radicals and Generalized Eigenspaces

I am currently reviewing Jordan Normal Form. Say we have $T,$ a linear operator, on a finite-dimensional vector space $V.$ So if we consider an eigenvector $v$ with eigenvalue $\lambda,$ then our ...
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Descending chain of reductions of an ideal

Let $(R, \mathcal{m},K)$ be a local ring and $J_0\supseteq J_1 \supseteq J_2\supseteq \cdots$ are all reductions of $I$. Then $\bigcap_{n\ge 0}J_n$ is also a reduction of $I$. I have done. I take ...
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Prime ideals of $R[x]$ that intersect $R$ in $P$

Let $R$ be a noetherian ring and $P$ a prime ideal of height $h$. Show that the prime ideals $Q\subset R[x]$ that intersect $R$ in $P$ are of the following two kinds, with height as shown: $Q=...
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An ascending chain of prime ideals.

I am trying to get used to $\operatorname{Spec}$ of a ring. I know an example, when one prime ideal is contained in another for $\mathbb{C}[x,y]$. $(f) \subset (x-a,y-b)$, where $f(a,b) = 0$. Is ...
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Find irreducible and connected components of $\operatorname{Spec}(\mathbb{C}[x] \times \mathbb{C}[y])$

Find irreducible and connected components of $\operatorname{Spec}(\mathbb{C}[x] \times \mathbb{C}[y])$ That's the problem 11.1 from commutative algebra course As answered here we can see the ...
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When a two-generated ideal in $\mathbb{Z}[\sqrt{-3}]$ is the unit ideal or principal?

I am trying to figure out when the ideal $(a_1+b_1\sqrt{-3},a_2+b_2\sqrt{-3}) $ is the unit ideal or principal in $\mathbb{Z}[\sqrt{-3}]$. Any hints?
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commutator ideal for direct sum

Is commutator ideal compatible with direct sum? Let's take $\mathfrak{sl}_2(\Bbb K)\oplus\mathfrak{sl}_2(\Bbb K)$ which is semi-simple because $\mathfrak{sl}_2(\Bbb K)$ is simple Lie algebra. So we ...
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Proving that a prime ideal is principal

Suppose $Q_1, Q_2\in \mathbb{C}[X_0,\dots,X_n]$ are irreducible homogeneous quadratic polynomials such that $V(Q_1, Q_2)$ is an irreducible projective variety of degree two and codimension two in $\...