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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Show that $k[x,y]/(xy-1)$ is not isomorphic to a polynomial ring in one variable. [duplicate]

Let $R=k[x,y]$ be a polynomial ring ($k$, of course, is a field). Show that $R/(xy-1)$ is not isomorphic to a polynomial ring in one variable. I can see that the polynomial $x+y$ is in ...
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+50

A Problem for Nil-Ideals

Consider a ring $R$ and $I$ be a finitely generated nil-ideal of $R$. Is $I$ a nilpotent ideal? I have proved this for commutative rings. But for non-commutative rings I think this may not be true. ...
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64 views

Writing down proof about graded ideal in multilinear algebra

I have a very simple question, but since this is the first time I'm dealing with graded ideals and so on it seems more difficult than it really is. Suppose $V$ is a finite dimensional vector space ...
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51 views

Maximal ideal of polynomial ring over a subfield

Let $L/K$ be an algebraic extension of fields. Let $B = L[X,Y]$ and $A = K[X,Y]$. Suppose $a$, $b \in L$ and $m = (X-a,Y-b)$ is an ideal of $B$. Show that $m$ and $m \cap A$ are maximal ideals of ...
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+50

Proving the Ideal Generated by the Coefficients of $f(X)\cdot g(X)\in R[X]$ is $R$.

Let $R$ be a commutative ring with unity, and let $f(X),g(X)\in R[X]$. Assume the ideals generated by the coefficients of $f(X),g(X)$ are both $R$. Prove that the ideal generated by the ...
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1answer
38 views

Is it true that $I\cap J\subset IJ\subset I+J\subset I\cup J$

Is it true that $I\cap J\subset IJ\subset I+J\subset I\cup J$ If $R$ is a commutative ring and $I,J$ are any ideals of $R$, I don't know how the product is usually defined but I think for $IJ$ is ...
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40 views

Prove by definition that $(x,2)\subset\mathbb Z[x]$ is a maximal ideal

When the polynomial ring $\mathbb{Z}[x]$ is quotiented by the ideal $(2,x)$ we get a field as $\mathbb{Z}[x]/(x,2)\cong\mathbb{Z}/(2)\cong\mathbb{Z}_{2}$ which is a field. But I ...
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47 views

Ideals in a ring as geometric objects?

I am interested in learing about the possibility of (one-sided) ideals in a ring being repreented geometrically. In other words, about their status as geometric objects (after all, they can be dealt ...
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102 views

Maps to quotient rings

If $R$ is a ring, $\mathfrak{a}$ is an ideal of $R$ and $S=A[x,y,z,\dots]$ where $A$ is a commutative ring, then is there a ring $S^\prime$ such that there is a bijection: ...
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24 views

Composition of module homomorphisms and their kernels

Let $R$ be a principal ideal ring and let $I$ and $J$ be two ideals of $R$. Suppose $\phi: R \times R \rightarrow R \times R$ and $\psi: R \times R \rightarrow R \times R$ are two $R$-module ...
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Inverse images of ideals

I was trying to solve the following exercise: Let $f\colon R\to S$ be a ring epimorphism, $I \subseteq S$ be an ideal, and $J = f^{-1}(I)$. Check that if $I$ is maximal (resp. prime) then $J$ is ...
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For a prime integer $p \in \{2, 3, 5, \cdots\}$, is $pR$ a maximal ideal in $R$?

If $R$ is a commutative ring with unit and $p$ is a prime number ($2,3,5,\cdots$), then is $pR$ a maximal ideal of $R$? If not what conditions should I impose on $R$?
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Show that $\langle x,y\rangle$ is not projective as a $\mathbb{Q}[x,y]$ -module. [closed]

I took this exercise for a long time but I can't prove it. Show that $\langle x,y\rangle$ is not projective as a $\mathbb{Q}[x,y]$ -module. Anyone could help me?
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2answers
188 views

Can every ideal have a minimal generating set?

Let $I$ be an ideal of commutative ring $A$ with unity. Does $I$ have a minimal generating set? At times, I am able to compute what they are for specific example, but it seems like it is true in ...
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1answer
30 views

Modify this formula : $R/I \cong \phi[R]/\phi[I]$

Let $R$ be a ring and $I$ an ideal of $R$, and let $\phi : R\longrightarrow R'$ be a ring homomorphism. Studying by myself, I have a conjecture the following: $$R/I \cong \phi[R]/\phi[I].$$ This ...
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22 views

Module endomorphisms with the same kernel

Let $R$ be a finite commutative principal ideal ring. Let $n$ be a positive integer. For $i=1, \ldots, n-1$ we let \begin{align*} w_i := w_i(x_{i+1}, \ldots, x_n) = \sum_{j=i+1}^n t_{ij}x_j \in ...
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91 views

finitely generated ideal and number of generators

let $I$ be a finitely generated ideal of ring $R$. Suppose $I/I^2$ as an $R/I$ module is generated by $r$ elements then question is to prove that $I$ is generated by $r+1$ elements.. I have tried ...
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58 views

How to find prime ideals of $3\Bbb Z$?

Which one of these is prime of $3\Bbb Z$? $42\Bbb Z$,$24\Bbb Z$,$12\Bbb Z$,$9\Bbb Z$ and $33\Bbb Z$ I tried to check their factor groups if they are integral domains. because An ideal I in a ...
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1answer
38 views

Every prime ideal is maximal [duplicate]

Problem: Show that if R is a finite ring, then every prime ideal of R is maximal. My attempt: Let I be a prime ideal of R. Then, by definition of a prime ideal, ab ∈ I implies a ∈ I or b ∈ I for ...
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1answer
45 views

Is radical of finitely generated ideal finitely generated?

Let $R$ be non-noetherian commutative ring with identity and $I$ be a finitely generated ideal of $R$; say $I = (a_1, \cdots, a_n)$. Question.1 Is $\sqrt I$ necessarily finitely generated? ...
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Isomorphism of modules + tensor product

Is it true that: $$M{\otimes}_{A}(A/I) \cong M/IM$$ and $$IM \cong I {\otimes}_AM$$ where $A$ is a commutative ring, $M$ an $A$-module, and $I \subset A$ an ideal.
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Ideals in direct product of algebras over a field [duplicate]

Let $ B_1,...,B_n$ be $k$-algebras ($k$ is a field), $ B=\prod_{i=1}^{n}B_i $ their direct product, and $ J_i$ an ideal of $B_i$. I must to prove that the direct product $ J=\prod_{i=1}^{n}J_i $ ...
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Every ideal has a FFR

Let $A$ be a regular local ring. Then every ideal has a finite free resolution. My thoughts: it's easy to prove that every ideal $I$ has a free resolution. In fact $I$ is finite and there is a ...
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1answer
50 views

$M=I\times J $ for some $I,J$ [duplicate]

Let $R,S$ be two rings with identity. Prove that every ideal of $R\times S$ is of the form $I \times J$ where $I$ is an ideal of $R$ and $J$ is an ideal of $S$ . Obviously $I \times J$ is an ideal of ...
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1answer
341 views

Ideals in direct product of rings

I am trying to solve this problem: Let $ R_1,...,R_n$ be rings with identity. Every ideal of $R=\prod_{i=1}^n R_i$ is of the form $\prod_{i=1}^n I_i$ where $ I_i$ is an ideal of $R_i$. The ...
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2answers
79 views

Question on prime ideals of ${\mathbb Z}[x]$

I was thinking this question myself: Consider the topological space $(\text{Spec}(\mathbb{Z}[x]),T )$ where open sets $D_I$ in $T$ are given as indexed by ideals $I$ in $\mathbb{Z}[x]: \; D_I =\{p\in ...
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1answer
42 views

How to show that an ideal is maximal

How do you show that $\langle y^2+2, x-1 \rangle$ is a maximal ideal in $\Bbb Q[x,y]$? I know that if you add another element that is not in this ideal, you should get the whole ring, thus ...
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56 views

Solving the Mordell Equation $y^2 = x^3 − 2$; what would be a general strategy?

I am looking at the solution provided in my lecture notes for solving this particular mordell equation: $$y^2 = x^3 − 2$$ which factors into: $$ (y- \sqrt {-2})(y+ \sqrt {-2}) = x^3 $$ In the ...
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Proof review: Every maximal ideal of ring of continuous functions has the same form

Let $R$ be the ring of real-valued continuous functions on $[0,1]$. If $M$ is a maximal ideal of $R$ prove $\exists \lambda \in [0,1]$ s.t. $M = \{f(x) \in R : f(\lambda) = 0 \}$. (from Herstein ...
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Checking whether $x^2-5$ is prime but not maximal

I want to find an example of a prime ideal that is not maximal. I thought about $x^2-5$. We know that $Z[\sqrt{-5}]\cong Z[x]/(x^2-5)$ is an integral domain, therefore is $x^2-5$ prime. However, I ...
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Converse of Chinese Remainder Theorem

Chinese Remainder Theorem for commutative rings with identity Let $R$ be a commutative ring with identity. If $I, J$ are ideals of $R$ satisfying $I+J=R$, then there is an isomorphism of rings: ...
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A question on $R$-modules

Let $M$ be a non-trivial irreducible (simple) $R$-module . Let $0 \ne m \in M$ and $A(m_0):=\{x \in R: xm_0=0\}$ , then is $A(m_0)$ a maximal left-ideal of $R$ and as $R$-modules , $M$ and $R/A(m_0)$ ...
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calculating height of a kernel [closed]

Consider the map $\phi: K[X_1,X_2,X_3,X_4] \to K[T,U]$ given by $X_1 \to T^4$, $X_2 \to T^3U$, $X_3 \to TU^3$, $X_4 \to U^4$. The question asks to show $\operatorname{height}(\ker(\phi))=2$. Can we ...
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About Relatively Prime Ideals

I am a physics Master student who has been studying abstract algebra by himself. I have two questions about relatively prime ideals. My first question goes as follows: Let $R$ be a ring, and ...
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Prove that $I^2$ is principal.

Consider the ideal $I=(2,\sqrt{-10})$ of $\mathbb{Z}[\sqrt{-10}]$. Prove that $I^2$ is principal. My Try: $I^2=(4,-10,2\sqrt{-10})$. I tried to prove that $I^2=(\sqrt{-10})$. But failed. Is my ...
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Can MAGMA write Groebner basis elements in terms of the original generators?

Consider the free algebra $F = \mathbb Q(a)\langle x, y, z\rangle$ and the ideal $$I = \langle xy - ayx, yz - zy, xz - zx - y\rangle$$ According to the following code $y^2 \in I$. ...
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2answers
39 views

For $I,J$ ideals $P$ Prime ideal, show that $IJ\subset P \iff I\cap J \subset P$

Question : Prove the following equivalence $IJ\subset P \iff I\cap J \subset P \iff$ $I$ or $J \subset P$ I was able to do this $IJ \subset I$ and $IJ \subset J$ so $IJ \subset P$ $IJ \subset I$ ...
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Confusion regarding a statement in Atiyah-Macdonald

Atiyah-Macdonald says the following: If the ideals $a_i, a_j $ are co prime, then $\Pi a_i= \cap a_i $ What does this even mean? For example, we know that $(2), (3) $ are co prime in the ring ...
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Is $(xy-1)$ a maximal ideal in $\mathbb C[x,y]$?

I learnd that the maximal ideals in $\mathbb C[x,y]$ have the form $(x-z_1, y-z_2)$ by the Nullstellensatz. But if we set $I=(xy-1)$ then $\mathbb C[x,y]/I$ is isomorphic to $\mathbb C[x,1/x]$ which ...
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164 views

Show that the trace class operators on a Hilbert space form an ideal

Let $(H, (\cdot, \cdot))$ be a separable Hilbert space over $\mathbb{L} = \mathbb{R}$ or $\mathbb{C}$. Suppose that $\{\phi_n\}_{n=1}^\infty$ is an orthonormal basis for $H$. Let $\mathcal{B}(H)$ ...
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1answer
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Expressing polynomial as linear combinaion

I found these questions in Adams Introduction to Groebner bases Let $f=x^6-1$ and $g=x^4+2x^3+2x^2-2x-3$. Let $I=\langle f,g\rangle$. Calculate the polynomial that generates $I$ alone. After a ...
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1answer
61 views

Riemann-Roch Theorem and Ideals of a Ring

I found in some Math book a comment stating that the study of Ideals in ring theory à la Dedekind (all kinds of ideals? only one-sided ideals?) could be transferred to other areas (specifically, ...
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Some doubts about right ideals of a ring

I would like to know whether the following paragraph regarding right ideals and modules is correct. Any comment or help is welcome: A right ideal of $R$ is just a submodule of the right $R$-module ...
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1answer
48 views

Proof about the difference between right and left ideals in a ring

I have tried get a version of the proof stating that a left ideals of a ring is not, in general, a right ideal, and viceversa. Is my formulation right? Comments and corrections are welcome. I have ...
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$R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$

$R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$ My brief attempt to try use Bezout theorem at a PID. but unsuccess.. Thanks any help.
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1answer
90 views

How to show that $\mathbb{C}[x_1,x_2,x_3, x_4]/(x_1x_2 - x_3x_4, x_1x_3 - x_2x_4, x_1x_4 - x_2x_3)$ is an integral domain?

I am looking for a way to show that the ring $\mathbb{C}[x_1,x_2,x_3, x_4]/I$ where $$I = (x_1x_2 - x_3x_4, x_1x_3 - x_2x_4, x_1x_4 - x_2x_3)$$ is an integral domain. In other words I want to show ...
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2answers
71 views

Prove that an ideal $ \mathfrak{m} $ of a commutative ring $ R $ is maximal iff $ R/\mathfrak{m} $ is simple.

Could someone give me a hint on whether I’m on the right track or not? For sufficiency, I tried the following: Suppose that $ \mathfrak{m} $ is a maximal ideal. With the quotient map, we get $ ...
2
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1answer
65 views

How does one find the Zariski closure of a set?

I've started to learn algebraic geometry this week (so I do not have much knowledge in the subjet) and, after reading the definition of the Zariski closure $V(I(S))$ of a set $S$, I've tried to do the ...
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1answer
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$I\otimes I$ is torsion free for a principal ideal $I$ in domain $R$

Question is : Suppose $I$ is a principal ideal in a domain $R$. Prove that the $R$ module $I\otimes_R I$ is torsion free. Suppose we have $r(m\otimes n)=0$.. Just for simplicity assume that ...