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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The rings Z[$\sqrt{6}$] and Z[$\sqrt{7}$] are PIDs. Exhibit generators for their ideals (3,$\sqrt{6}$), (5, 4 + $\sqrt{6}$), (2, 1 + $\sqrt{7}$)

The rings Z[$\sqrt{6}$] and Z[$\sqrt{7}$] are PIDs. Exhibit generators for their ideals (3,$\sqrt{6}$), (5, 4 + $\sqrt{6}$), (2, 1 + $\sqrt{7}$) Can I get walked through one of them so that I ...
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Chinese Remainder Theorem proof using maximal ideals

I am trying to prove Generalized Chinese Remainder Theorem based on the following hint. The statement is: $\{Q_i\}$ is a set of coprime ideals of a commutative ring $R$ and $\phi : R \to \prod_{i} ...
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How to show that $J$ is a left ideal of $R$?

Let $R$ be a ring and $x \in R$. Let, $J = \{ax|a \in R\}$ Show that $J$ is a left ideal of $R$. To show something is a left Ideal(or a right ideal), do we need to show that it is a subgroup first? ...
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88 views

Showing that if the initial ideal of I is radical, then I is radical.

I need to show that, given a term order $<$ and an ideal $I$, if $in_<(I)$ is radical, then $I$ is radical. Any help or hints would be appreciated as I'm not really sure where to start, since I ...
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37 views

Inclusion of fractional ideals implies equality

Let $R$ be a integral domain and let $\mathfrak U\subseteq\mathfrak B$ two ideals of $R$ such that $\mathfrak UR_\mathfrak p=\mathfrak BR_\mathfrak p$ for all maximal ideals. Then $\mathfrak ...
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52 views

Ideal of $\mathbb{C}[X,Y]$ contained in infinitely many distinct proper ideals

Let $R=\mathbb{C}[X,Y]$, the polynomial ring in two variables over $\mathbb{C}$, and consider the (principal) ideal $I=(X^3-Y^2)$ of $R$. I've shown that $I$ is a prime ideal and that it is not ...
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Show that a collection of functions $I$ is an ideal

Let $R$ be the ring of all continuous function on $[0,1]$, and let $I$ be the collection of functions $f(x)$ in $R$ with $f(1/3)=f(1/2)=0$. Prove that $I$ is an ideal of $R$ but not a prime ideal. I ...
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49 views

A proof about prime ideals

Assume $R$ is commutative. Prove that if $P$ is a prime ideal of $R$ and $P$ contains no zero-divisors then $R$ is an integral domain. Proof: let $ab \in P$ where $ab \not= 0$. that means $a \in P$ ...
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28 views

The ideals of $A \times B$ for fields $A,B$.are principal

Let $A$, $B$ fields. I showed that $A\times B$ is ring which is not field. I need to show that every ideal in $A\times B$ is principal. Let $I$ ideal of $A$ and $J$ ideal of $B$. iF $I\times ...
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Show that $\langle 13 \rangle$ is a prime ideal in $\mathbb{Z[\sqrt{-5}]}$

To show that $\langle 13 \rangle$ is a prime ideal in $D= \mathbb{Z[\sqrt{-5}]}$, I could show that $13$ is an irreducible element of $D$ but as $D$ is not a U.F.D, it is not of much use I guess. How ...
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37 views

Extension of an ideal to a subring of the ring of fractions

Let $A$ be a domain, and $B$ an $A$-algebra inside $\text{Frac}(A)$. Let $x/y\in B$. Then $(yA:_Ax)B\neq B$ if and only if there is a prime ideal $\mathfrak{p}\in \text{Spec}(A)$ such that ...
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Help with Atiyah-McDonald proof. $1 + \hat{m} = $ units $\implies A$ is a local ring.

Prop 1.6 i) Let $A$ be a ring and $\hat{m} \neq (1)$ an ideal of $A$ such that every $x \in A - \hat{m}$ (set difference) is a unit in $A$. Then $A$ is a local ring and $\hat{m}$ its maximal ideal. ...
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33 views

$p$ is a prime in $\mathbb{Z}[i]$ if and only if $-1$ is not a square modulo $p$, direct proof using $\mathbb{Z}[x]$?

I'm trying to show directly that $p$ is prime in $\mathbb{Z}[i]$ if and only if $-1$ is not a square modulo $p$, using $\mathbb{Z}[x]$. I see how to prove the result using the description of prime ...
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22 views

Show that $\mathscr{C}_{X,a}\cong\mathscr{C}_{A^n,a}/ I(X)\mathscr{C}_{A^n,a}$.

Let $X\subset A^n$ be an affine variety, and let $a\in X$ be a point. Show that $\mathscr{C}_{X,a}\cong\mathscr{C}_{A^n,a}/ I(X)\mathscr{C}_{A^n,a}$, where $I(X)\mathscr{C}_{A^n,a}$ denotes the ideal ...
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44 views

If $R=K[X]/(X^n)$, can represent any element as polynomial with degree $<n$

Let $K$ be a field and $R=K[X]/(X^n)$ where $n \in \mathbb{Z}_{n\geq1}$ and $(X^n)$ is the ideal generated by $X^n$. We denote $x:=X+(X^n) \in R$, any equivalence class $r$ in $R$ has a representing ...
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18 views

M an R-module, where M is a commutative ring, if $M≅R/I$ for some ideal I of R, then $M$ is a cyclic R-module

M is an R-module, where M is a commutative ring, if $M≅R/I$ for some ideal I of R, then show $M$ is a cyclic R-module. Note: $M$ is a cyclic R-module means $M=<m>$ for some $m\in M$. Please ...
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Inclusion of subring in Ideal

Let $K$ be a commutative ring with mutpilicative identity and $m \ge 3$. Let $L(m,K)$ be a subring of Lie ring of matrices with coefficients from $K$ and traces = $0$: $ \{ (a_{ij}) \in M_m (K) | ...
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Show $(⊕M_α)/I(⊕M_α)\simeq ⊕(M_α/IM_α)$ for ${M_α}$ $R$-modules, and $I$ is an ideal of the commutative ring $R$

Let $(M_α)$ be a collection of $R$-modules, and $I$ is an ideal of the commutative ring $R$. Show that $(⊕M_α)/I(⊕M_α)$ is isomorphic to $⊕(M_α/IM_α)$ as $R/I$-modules. Please help, thanks a lot!
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how to define the map between the isomophism $(⊕Mα)/I(⊕Mα)\simeq ⊕(Mα/IMα)$?

May I ask how to define the map between the isomorphism $(⊕M_α)/I(⊕M_α)\simeq ⊕(M_α/IM_α)$?. where ${M_α}$ is a collection of $R$-modules, and $I$ is an ideal of $R$. we know $(⊕M_α)/I(⊕M_α)\simeq ...
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Proof that ideal in Lie ring can be represented as sum of 2 Lie subrings

Let $K$ be a commutative ring and $m \ge 3$. Let $L(m,K)$ be a Lie subring of matrices with coefficients from ring $K$ that contains matrices with null traces, $L(m,K)=\{(a_{ij}) \in M_m(K) | ...
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23 views

A left ideal need not be a right $R$ module

Let $R$ be a ring, it naturally follows that every left ideal of $R$ will also be a left $R$ module and an analogous result will also hold for right ideals of $R$. I was trying to show that a left ...
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46 views

Image of ideal under the isomorphism given by the Chinese Remainder Theorem.

Suppose that $\mathfrak{p}_1,\ldots,\mathfrak{p}_n$ are maximal ideals of a ring $R$. Then $\mathfrak{p}_i+\mathfrak{p}_j=R$ with $i\neq j$ and $\mathfrak{p}_i^a+\mathfrak{p}_j^b=R$ with $a,b$ ...
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Intersection of ideals in the ring of formal power series

Let $R$ be a commutative ring and $I,J$ ideals in $R$. Denote by $R[[X]]$ the ring of formal power series with coefficients in $R$. If $A\subseteq R$, denote by $A^e$ the ideal in $R[[X]]$ generated ...
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26 views

Show that $J=I(S^{-1}R)$

Let $R$ be a commutative ring, $S\subset R$ a multiplicative set, and $J\subset S^{-1}R$ an ideal. Let us define $I:=\{x\in R:\frac x1\in J\}$. Show that $J=I(S^{-1}R)$ We had the homomorphism, ...
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Why is $(2, 1+\sqrt{-5})$ not principal?

Why is $(2, 1+\sqrt{-5})$ not principal in $\mathbf Z[\sqrt{-5}]$? Say $(2,1+\sqrt{-5})=(\alpha)$, then since $2\in(2,1+\sqrt{-5})$ we have $2\in (\alpha)$, so $\alpha\mid2$ in $\mathbb ...
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51 views

Rings with same quotient field

Let $R$ be an integral domain and $0 \neq I$ an ideal of $R$. Denote by $\phi: R \rightarrow R/I$ the canonical homomorphism. Let $S$ be a subring of $R/I$ such that $R/I$ is integral over $S$. ...
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Some questions about Milne's algebraic geometry notes.

I'm trying to go through the details of an example in J. Milne's algebraic geometry notes (p.42, 2.22). He gives us the general fact that for two algebraic subsets $W,W'\subseteq K^n$, we have ...
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36 views

Showing that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal, and also finding its cardinality?

How to show that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal? Is it possible to find the cardinality of $\mathbb{Z}[i]/I$ as well? I know how to show that it is an integral ...
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46 views

Determining whether a ring is a principal ideal ring or not

I have been attempting to attack the following problem off and on for a few weeks now, without much success: Is the ring $R=\mathbb{Z}_{4}[x]$ of polynomials with coefficients in $\mathbb{Z}_{4}$ ...
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68 views

When two ideals are coprime? [closed]

Let $K = \mathbb Q{(\sqrt{-m})}$ be an imaginary quadratic number field. Let $ \alpha = a + b \sqrt{-m}$ be an algebraic integer, with $\mbox{gcd}(a, b) = 1$. Then how to show that ideals $\langle ...
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41 views

what can be said about $spec(R_m)$, where $R_m$ is localization of $R$ at maximal ideal $m$

I've seen how if $p$ is a prime ideal of $R$ and $R_p$ is the localization of $R$ at $P$, then $P_p$ is the unique maximal ideal of $R_p$, but what if we had a maximal ideal $m$ of $R$, then $R_m$ ...
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Trouble understanding Eisenbud Exercise 2.19a

I'm working through the "Commutative algebra with a view toward algebraic geometry" book and stumbled onto an exercise I'm struggling to answer. Let $R$ be a ring and let $M$ be an $R$-module. ...
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36 views

Prove the union of ideals is an ideal

prove that $I_{1} \subseteq I_{2} \subseteq I_{3} \subseteq....$ are ideals of $R$ then $\bigcup_{n =1} I_n$ is an ideal of $R$. I am having a hard time picture this in my mind. Help anyone.
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114 views

Ring with maximal ideal not containing a specific expression

Main question : May there exist an integral domain $R$, with fraction field $K$, that fulfills the following condition: there exists $x\in K$, $x\not \in R$ and a maximal ideal $\frak m$ of $R[x]$, ...
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How to compute the prime ideal factorization of a given ideal in an algebraic ring

I have been working on a problem involving integral ideals in algebraic ring $\mathcal{O}_K$. And it involves the unique factorization of a integral ideal ${I}$ into product of powers of some prime ...
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62 views

Converse to Chinese Remainder Theorem

So as seen on this question Converse of the Chinese Remainder Theorem, we know that if $(n,m) \neq 1$, then $\mathbb{Z} /mn \mathbb{Z} \ncong \mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z}$, ...
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Symmetry of two-sided ideals

I was thinking about two-sided ideals and have some intuition-guided, soft questions regarding them. Since I don't have anyone to talk to about such subject matter, I thought I'd ask. Let a be an ...
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1answer
54 views

On the polynomial ring $ \mathbb{R}[x,y] $, and the sine and cosine functions.

I was investigating relationships between commutative algebra and real analysis when the following problem came into mind. Problem. Let $ P \in \mathbb{R}[x,y] $. If $ P(\sin(\theta),\cos(\theta)) ...
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28 views

Question about repeated quotient groups

I was wondering if there was a simpler representation for the quotient group $(K[x,y]/\langle xy\rangle)/\langle x, y-1\rangle$. Where $K[x,y]$ is the polynomial ring with variables $x,y$ over the ...
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21 views

What is $dim_\mathbb{F} (R/I)$ if $R/I$ is a graded ring and $I$ is homogeneous?

If $I$ is a homogeneous ideal, when is $R/I$ a graded ring? And if $R/I$ is a graded ring and we think of it as a $\mathbb{F}$-vector space, is the $dim_{\mathbb{F}} (R/I)$ always finite and if it is ...
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R-Module Homomorphism from $I\otimes I$ to $\mathbb{Z}_2$, where $I = (2,x)$

Hello everyone, can someone tell me how to prove part(c).This is my atempt: Define $\Phi: I\otimes I \longrightarrow \mathbb{Z}_2$ by $\Phi(p(x) \otimes q(x))= \frac{p(0)}{2}*q'(0)$, where $q'$ ...
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every z°-ideal I in C(X) is a rez°-ideal

I need to show that every z°-ideal I in C(X) is a rez°-ideal? in order to get a rez°-ideal I need to find an ideal an ideal J in C(X) such that I⊊J and I is a 〖z°〗_J-ideal. shall anybody help me in ...
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35 views

Show if $R$ is Noetherian, then $R_S$ is Noetherian [duplicate]

Show if $R$ is Noetherian, then $R_S$ is Noetherian. here is what I have read from somewhere else. Suppose $R$ is Noetherian and $J$ is an ideal $R_S$. Then $J=IR_S$ for some ideal $I$ of $R$. Since ...
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16 views

Show if $P$ is minimal prime ideal of $R$ then $PR_P$ is the only prime ideal of $R_P$

SHow if $P$ is minimal prime ideal of $R$ then $PR_P$ is the only prime ideal of $R_P$. Here are what I know and don't need to prove: I know $PR_P$ is the maximal ideal of $R_P$. I know $R_P$ is a ...
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2answers
67 views

Finding Radical of an Ideal [duplicate]

Given the ideal $J^\prime=\langle xy,xz-yz\rangle$, find it's radical. I know that the ideal $\langle xy,yz,zx\rangle$ is radical ideal but that's not the case. How can I compute the radical ...
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19 views

show if $P$ is minimal prime ideal of $R$ then every element of $PR_P$ is nilpotent.

Show if $P$ is minimal prime ideal of $R$ then every element of $PR_P$ is nilpotent. The only idea that I come to mind is, we know $PR_P$ is the maximal ideal of $R_P$. Since $P$ is a prime ideal of ...
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1answer
18 views

Coprime ideal definition

I am learning about ideals in my algebra class. If I have a ring $R$, I know that two ideals $I$ and $J$ in $R$ are coprime if $I+J=R$. I also know that $\mathbb{Z}$ is a principal ideal domain. I was ...
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1answer
40 views

Abstract Algebra-Quotient Rings [duplicate]

I am having trouble understanding ideals. If $R$ is a commutative ring, and $I$ an ideal of $R$, can someone please explain what an ideal of $R/I$ looks like? Thank you.
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39 views

If $Q$ is a radical ideal of $R$ with $Q∩S=∅$, then $QR_S$ is a radical ideal of $R_S$

If $Q$ is a radical ideal of $R$ with $Q∩S=∅$, show $QR_S$ is a radical ideal of $R_S$. Here is what I have done, since $Q$ is radical, let $q\in Q$ , then $q^n\in I$ for some $I$ ideal. let $a\in ...
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1answer
20 views

Is the localization of R by S is a subset of the ring R

Let $S$ be a multiplicatively closed subset of a commutative ring $R$. Then is it true that the localization $R_S=\{r/s:r\in R, s\in S\}$ a sub-ring of $R$? I think it is true, because $r/s=rs^{-1}$ ...