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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Can we use the Nullstellensatz?

In $\mathbb{C}[x, y, z]$ we have that $V=\{y-x^2, z-x^3)=\{(t, t^2, t^3) | t \in \mathbb{C}\}$. To show that $$I(V(y-x^2, z-x^3))=\langle y-x^2, z-x^3\rangle $$ can we use the Nullstellensatz?? ...
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26 views

Why is that the radical ideal?

In my lecture notes we have the following: Definition: $f, g \in \mathbb{C}[x, y]$ $f \sim g \Leftrightarrow \exists c \in \mathbb{C}, c \neq 0$ such that $g=cf$ Example: If $f \sim g ...
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$V_1=V(x-y)$ and $V_2=V(x+y)$ are algebraic sets

I am looking at irreducible algebraic sets. $V \subseteq K^n$ is an algebraic set $\Leftrightarrow$ it is of the form $V(I)$, where $I=$radical Ideal of $K[x_1, x_2, \dots , x_n]$. At my lecture ...
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14 views

Algebraic set - Radical Ideal - $Rad(Rad(I))=Rad(I)$

In my lecture notes we have the following: $V \subseteq K^n$ is an algebraic set $\Leftrightarrow$ it is of the form $V(I)$, where $I=$radical Ideal of $K[x_1, x_2, \dots , x_n]$. It stands that ...
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1answer
21 views

Algebraic Set-Radical Ideal-Nullstellensatz

In my lecture notes there is the following: $$I \rightarrow V(I) \rightarrow I(V(I))$$ It stands that in general $I \subsetneq I(V(I))$. The equality stands if and only if $I$ is a radical ...
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23 views

A question about left ideals

Let $V$ be a finite-dimensional vector space over a field $F$. I need to prove that for every left ideal $I$ of $\operatorname{End}_F(V)$, there is only one subspace $W$ of $V$ for which $I$ = {$A ...
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15 views

Proof of the proposition $V(S)=V(\langle S \rangle )$

In my lecture notes we have the following: Proposition: $$V(S)=V(\langle S \rangle )$$ Proof: $$\langle S \rangle=\left \{\sum_{i=1}^m g_i f_i | f_i \in S, g_i \in R=K[x_1, x_2, \dots , ...
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22 views

Operations with ideals: sum and product

Operations at ideals. The sum is defined as $$I_1 + I_2 + \dots + I_m =\{a_1+a_2+\cdots +a_m\mid a_i \in I_i\}.$$ It can be proven that $$I_1 + I_2 + \dots + I_m \trianglelefteq R$$ and each ...
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$Rad(I)$ is an ideal of $I$

$$Rad(I)=\{a \in R | \exists n \in \mathbb{N} \text{ such that } a^n \in I\}$$ R is a commutative ring, I is an ideal. To show that $Rad(I)$ is an ideal of $I$, we have to show that for $a,b \in ...
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22 views

For a maximal ideal $M$ of $R$ of a commutative ring $R$ ( not necessarily with unity ) , then is $R/M$ a simple ring ?

Let $M$ be a maximal ideal of a commutative ring $R$ ( not necessarily with unity ) ; then is it true that the only ideals of $R/M$ are the trivial ones i.e. is it true that $R/M$ is a simple ring ? ...
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14 views

No such prime ideal contains $I_1+I_2\implies I_1 $ and $I_2$ are relatively prime

It's clear to me that if $I_1$ and $I_2$ are two relatively prime ideals of a ring $R$, then there is no such prime ideal containing $I_1+I_2$, since by definition of relatively prime ideals ...
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21 views

Two ideals that agree in the completion [on hold]

Suppose that $R$ is a Noetherian local ring with maximal ideal $\mathfrak m$, and that $I$ and $J$ are two ideals in $R$ with $\hat{I} = \hat{J}$ in the completion of $R$ at $\mathfrak m$. What can ...
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36 views

$I$ is maximal ideal $\implies$ $R/I$ has no proper ideals

I'm reading through a proof in a book on commutative algebra and in the proof it uses the fact that $I$ is a maximal ideal $\implies$ $R/I$ has no proper ideals, by using the correspondence theorem. ...
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77 views

$\mathbb{Z}_{(2)}$ has one maximal ideal

My lecture notes state that the set $\mathbb{Z}_{(2)}$, defined as $$\mathbb{Z}_{(2)}:=\left\{\frac{a}{b}\in\mathbb{Q}\mathrel{}\middle|\mathrel{}\gcd(a,b)=1\text{ and } 2\nmid b\right\}$$ has a ...
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32 views

Simplify $(y-x^2)\cap(y^2+2y+x^2)$

In the book "Commutative Algebra with a View Toward Algebraic Geometry (Eisenbud, 1995), exercise 1.10 one has to find the ring associated to the union of the circle $C:(y+1)^2+x^2=1$ and the parabola ...
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A relation between the Jacobson radicals of a ring and those of a certain quotient ring

Let $R$ be a ring $J(R)$ the Jacobson radical of $R$ which we define for this problem to be all the maximal left ideals of $R.$ I'm trying to prove the following proposition with only the definition ...
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1answer
40 views

$\sqrt{I}+\sqrt{J}=R$ implies $I+J=R$

Let $R$ be a commutative ring with unity and $I,J$ ideals of $R$. Suppose that $$ \sqrt{I}+\sqrt{J}=R $$ I want to show that this implies $I+J=R$. Take $r\in R$, then I can write $$ r=a+b, $$ for ...
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1answer
36 views

Quotient of ring is flat gives an identity of ideals

I have problem to understand and solve the exercise 1.2.14 on Qing Liu's book "Algebraic Geometry and Arithmetic Curves". It goes as follows: Let $A\to B$ be a ring homomorphism, and let $J$ be an ...
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47 views

Why prime avoidance lemma allows only at most 2 non-prime ideals?

Why prime avoidance lemma allows only at most 2 non-prime ideals? The following is the last part of the proof taken from wikipedia: For the case $n > 2$, choose $z_i \in E \cap (I_i - \cup_{j ...
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Algebra A over a field F contains no non-trivial left F-ideals if and only if A contains no non-trivial right F-ideals [closed]

Algebra $A$ over a field $F$ contains no non-trivial left $F$-ideals if and only if $A$ contains no non-trivial right $F$-ideals. Why this fact is true? Or is it true? I think it's easy thing, ...
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43 views

How to solve this algebra problem?

Let $e$ be the idempotent element of the ring R. If $\langle e\rangle$ is the principal ideal generated with $e$, show that $R\simeq\langle e\rangle\times A(\{e\})$. I think $A$ s ring which contains ...
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31 views

I need help to solve this problem

Let $R$ be a subring of a field $F$ such that for each $x \in F$ either $x\in R$ or $x^{-1} \in R$. Prove that if $I$ and $J$ are two ideals of $R$, then either $I \subseteq J$ or $J \subseteq I$.
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1answer
38 views

Prove that $(2)$ is a prime ideal in $\mathbb Z[w]$

Let $w\in\mathbb C$ be such that $w^3=1$ and $w\neq1$. Prove that $(2)$ is a prime ideal in $\mathbb Z[w]$, and describe $\mathbb Z[w]/(2)$. What I wanted to do is to show that $\mathbb Z[w]$ is a ...
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1answer
41 views

What happens if we change the definition of quotient ring to the one that does not have ideal restriction?

From Wikipedia: Given a ring R and a two-sided ideal I in R, we may define an equivalence relation ~ on R as follows: a ~ b if and only if a − b is in I. ...
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22 views

what happens if we adjoin elements in a ring not by ideals and quotient ring? [closed]

We often adjoin elements in a ring by using ideals which results in a quotient ring. What happens if we adjoin elements that cannot use ideals method? What is the general property of the resulting ...
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46 views

Why is $I$ often an ideal in quotient ring $A/I$?

When talking about quotient ring $A/I$, where $A$ is a ring, $I$ is often assumed to be an ideal. Why is this so? What makes ideals very important when discussing quotient ring?
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21 views

Showing the inverse map of a ring homomorphism of a prime ideal is again a prime ideal

Let $\phi : A \rightarrow B$ be a ring homomorphism and $I$ be a prime ideal of $B$. (i) Show that $\phi^{-1}(I)$ is a prime ideal of $A$, and (ii) find an example of $A$, $B$ and $I$ so that ...
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0answers
17 views

Direct sum of ideals over Dedekind domain [duplicate]

I'm trying to show that Let $\frak{a},\frak{b}$ be two ideals of a Dedekind domain $\cal{O}$. Show that there is an isomorphism \begin{equation*} ...
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1answer
40 views

For a commutative ring $R$, why does $1-ab$ being a non-unit leads to $1-ab \in M$ for some maximal ideal $M$?

Suppose there is a commutative ring $R$, without any restriction. Now suppose $a,b \in R$. If $1-ab$ is a non-unit, why is there at least one maximal ideal $M$ that $1-ab \in M$?
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1answer
37 views

Nullstellensatz: If $V(f)=V(g)$ we have that $Rad \langle f \rangle =Rad \langle g \rangle$

In my lecture notes I have the following: From the Nullstellensatz (NSS for short) we have the following: $$\text{ If } V(f)=V(g) \Rightarrow V(Rad(\langle f \rangle ))=V(Rad \langle g \rangle ) ...
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1answer
54 views

Sum of ideals-Intersection of algebraic sets

In my lecture notes I have the following: $$ \begin{array}{ccl} \text{Sum of ideals} & & \text{Intersection of algebraic sets} \\[4pt] I+J & \longrightarrow & V(I+J)=V(I)\cap V(J) \\ ...
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1answer
39 views

Flatness and intersection of ideals

This is Liu 1.2.6 a Let $B$ be a flat $A$-algebra. Show that for any finite family $\{I_\lambda\}_{\lambda\in \Lambda}$ of ideals of $A$, we have $\cap_{\lambda\in\Lambda}(I_\lambda ...
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42 views

Conditions on ideal b for fields or integral domains

Let $A$ be a ring and $b$ be an ideal of $A$. Prove that 1. $A/b$ is a field $\iff b$ is maximal 2. $A/b$ is an integral domain $\iff b$ is prime I figure that the first is derived from the fact ...
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1answer
31 views

The intersection of a and b is a superset of the product when a and b are ideals

Let a and b be ideals of a ring A. Define $$ab=\left\{{\sum_{j=1}^{n} a_jb_j|a_j\in a,b_j \in b,n \in \mathbb{N}}\right\}$$ Prove that $ab$ and $a\cap b$ are ideals of A, and that $a\cap b \supseteq ...
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19 views

Ideals of the quotient ring of A [duplicate]

Let A be a ring and b be an ideal of A. The quotient ring of A by b, denoted A/b is the ring of all equivalence classes A + b. Prove that the assignment $$c → c/b$$ induces a one-to-one ...
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26 views

If $I=\langle 12 \rangle$, then $Rad(I)=\langle 6\rangle$

To show that if $I=\langle 12 \rangle$, then $Rad(I)=\langle 6\rangle$, I did the following: $$36=3 \cdot 12 \\ 6^2=36 \in I \Rightarrow 6 \in Rad(I) \Rightarrow \langle 6 \rangle \subseteq Rad(I)$$ ...
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53 views

How to check if a polynomial is inside an ideal using a Groebner basis

I'm given that an ideal $I=\langle F_1, F_2, F_3, F_4, F_5, F_6, F_7\rangle$ $F_1=a+b+c-d-e-f$ $F_2=a+b+c-g-h-i$ $F_3=a+b+c-g-e-c$ $F_4=a+b+c-a-e-i$ $F_5=a+d+g-a-e-i$ $F_6=a+d+g-c-f-i$ ...
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60 views

Bijection between sets of ideals

Let $A$ be a ring and $\mathfrak{b}$ be an ideal of $A$. Prove that the assignment $$\mathfrak{c} \mapsto \mathfrak{c}/\mathfrak{b}$$ induces a one-to-one correspondence between the ideals of ...
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1answer
66 views

$M$ is maximal, $P$ is prime but not maximal

If $R$ is commutative with $1 \in R$, then each maximal ideal of $R$ is also a prime. The reverse doesn't hold. For example, $R=K[x, y], P=\langle x \rangle, M=\langle x, y\rangle$. Then ...
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1answer
48 views

Showing that for every monomial $x^u\in\operatorname{in}_{<}(I)$, there exists $f\in I$ s.t. $\operatorname{in}_<(f)=x^u$

Given an ideal $I\subset R=K[x_1, ...,x_n]$ and let $<$ be a term order on the ring $R$. I must show that $\forall x^u\in\operatorname{in}_<(I)$, $\exists f\in I$ s.t. ...
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1answer
64 views

Maximal ideals of $R[x_1,\ldots,x_n]$ that is $R$ is a commutative rings with identity

Let $R$ be a commutative ring with identity and $R[x_1,\ldots,x_n]$ a polynomial ring over $R$. What are maximal ideals in $R[x_1,\ldots,x_n]$? How are, if $R$ is a Hilbert ring (Jacobson ring)?
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1answer
43 views

Polynomial ring, prime ideal, factor ring

I want to prove that this ideal: $I=(y^3-xz, xy^2-z^2, x^2-yz)$ is prime in $K[x,y,z]$. I think it would be a good idea to prove that the factor ring $K[x,y,z]/I$ has no zero divisors. In this factor ...
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1answer
48 views

The monomials not inside $in_<(I)$ form a K-basis inside the Quotient ring

Given the quotient ring $T/I$, where $T=K[x_1,...,x_n]$ is a polynomial ring and $I$ is an ideal. I need to show that for any monomial $x^u:=x_1^{u_1}*...*x_n^{u_n}$, if the monomial is not inside ...
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1answer
48 views

Proving that S/I is a vector space

I'm given a polynomial ring $S=K[x_1,...,x_n]$ and $I$ is an ideal of $S$. I'm working on proving that the quotient ring $S/I$ is a vector spake over $K$. Since S is a ring, we already have some of ...
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1answer
82 views

Is the mentioned basis a Gröbner basis?

It's mentioned into my notes that if the ideal given as $I=\langle x+y+z, 3x-2y\rangle$, then $\{x+y+z, 5y+3z\}$ is a Gröbner basis for the ideal. I can see how $I=\langle x+y+z, 3x-2y\rangle=\langle ...
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1answer
33 views

fraction field of polynomial ring that is a finite extension of the base field

Let $k$ be a field. Let $P$ be a prime ideal of $k[x_1, ..., x_n]$. Let $K$ be a field of fractions of $k[x_1, ..., x_n]/P$. Suppose $K$ is a finite extension of $k$. Does it then follow that $P$ is ...
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1answer
22 views

$I+J=R$ and $r+s=1, r\in I,s\in J$ then $sx+ry\in IJ\Rightarrow x\in I$ and $y\in J$

Let $R$ be a commutative ring with unity. $I+J=R$ with $I,J$ Ideals and $r+s=1, r\in I,s\in J$ then $sx+ry\in IJ\Rightarrow x\in I$ and $y\in J$. It should be very obvious. How can I conclude that ...
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1answer
59 views

Correspondence principle applied to ideals of a quotient ring.

Let $I$ be an ideal of $R$. Prove that the ideals of $R/I$ are precisely of the form $J/I$ with $I \subseteq J$ and $J$ is an ideal of $R$. Can someone give me some hint on how to solve this problem? ...
1
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0answers
99 views

Prove that factor modules are isomorphic.

I'm trying to prove (from a previous post) that if $A=k[x,y,z]$ and $I=(x,y)(x,z)$ then $((x,y)/I)/((x,yz)/I) \cong A/(x,z)$. I did this by defining the homomorphism $\phi: A \to ...
0
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1answer
71 views

Prime ideals of infinite depth in Noetherian rings

I'm struggling with the definition of depth of prime ideals given in Atiyah's book: The depth of a prime ideal $p$ is longest strictly increasing chain of prime ideals starting at $p$. Clearly ...