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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absolutely convex ideals and $F$-spaces. [on hold]

I'm asked to prove this: $X$ is an [$F$-space] if and only if every ideal is [absolutely convex] I have a problem in proving the first direction that if $X$ is an $F$-space then every ideal is ...
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Isomorphism of quotient rings

In a course on Algebraic Number Theory, the lecturer says $$\mathcal{O}_K\cong \mathbb Z\left[\frac{1+\sqrt d}{2}\right] \cong\frac{\mathbb Z[x]}{\left( x^2-x-\frac{d-1}{4} \right)}$$ and so for a ...
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Gaussian Integers ring

I'm having troubles with my algebra homework. Could you please help me? Thanks. Let $\mathbb Z[i] =\{a+bi \mid a, b \in \mathbb Z\}$ be a Gaussian Integer set. 1) Show that ideal $I = (2+2i)$ is ...
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37 views

Understanding the ideal $IJ$ in $R$

I'm a little confused why our book defines an ideal $IJ$ in $R$ (where $I$ and $J$ are ideals) in such a complicated way: $$IJ=\left\{ \displaystyle\sum\limits_{i=1}^n a_i b_i\mid n\geq 1 ,a_i\in I, ...
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About the meaning of associated graded ideal

Let $G$ be any multiplicative group (abelian or not). Suppose that $R$ is a $G$-graded ring, i.e., there exists a family of additive subgroup $\{R_g\}_{g\in G}$ such that $R=\bigoplus_{g\in G}R_g$ ...
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Concerning ideals of $\mathbb Z[\sqrt m]$ and $\mathbb Z[\sqrt m] [x] $

For a given integer $m<-1$ or non-square integer $m>1$ , how do we calculate the quotient ring $\mathbb Z[\sqrt m]/I$ , for example its order or whether it is a field or has zero divisors or not ...
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39 views

Does validity of Bezout identity in integral domain implies the domain is PID ?

Let $D$ be an integral domain such that for any $a,b \in D$ , $Da+Db$ is a principal ideal , then must $D$ necessarily be a principal ideal domain i.e. should all the ideals of $D$ be principal ? ...
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Finitely generated ideal containing non finitely generated ideal

I've been thinking about the following Rotman's excercise, and just can't find an answer: Give an example of a commutative ring $R$ containing proper ideals $I\subsetneq J\subsetneq R$ with $J$ ...
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nilpotent right ideals

Theorem 3: Every nilpotent right (left) ideal is contained in a nilpotent two-sided ideal. Proof: Let $I$ be a nilpotent right ideal of $R$. By induction $(I + RI)^n ≤ I^n + RI^n$ for all ...
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A and B left ideals of ring R. Is $BA⊆A$?

Let $R$ be a ring. Let $A$ be left ideal of $R$, and $B$ be a left ideal of $R$. Is there any way I could show that $BA⊆A$? I was trying to use this fact to help me with another question, but I'm ...
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79 views

Unique factorization in prime ideals in a local ring

Let $R$ be a local commutative domain with maximal ideal $M$. Assume that every ideal of $R$ is a product of prime ideals in a unique way. I want to show that the only non-zero prime ideal of $R$ ...
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Proving an ideal is maximal

Let p be a prime. show that A = {(px,y) : x,y $\in$ $\mathbb Z$ } is a maximal ideal of $\mathbb Z$ x $\mathbb Z$. I am having trouble showing that A is maximal. To show A is an ideal, first note ...
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43 views

What is a linear combination, exactly?

I'm used to the definition of linear combination used in linear algebra textbooks. I'm reading the book Algebra by Artin and on page 357 he says: If $R$ is the ring $\mathbb{Z}[x]$ of integer ...
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45 views

Is $x \in X$ isolated iff the ideal $M_x := \{f \in C(X, \mathbb R) :f(x)=0\}$ of $C(X, \mathbb R)$ is principal ?

Let $(X,d)$ be a metric space , then is $x \in X$ isolated iff the ideal $M_x := \{f \in C(X, \mathbb R) :f(x)=0\}$ of $C(X, \mathbb R)$ is principal ? Do we need completeness of $X$ ?
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If $(X,d)$ is a finite metric space , then is every prime ideal of $C(X, \mathbb R)$ maximal?

If $(X,d)$ is a finite metric space , then is every prime ideal of $C(X, \mathbb R)$ maximal ? The thing is , since $X$ is finite , so it is compact , so ideal $M$ is maximal iff it is of the form ...
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If $(X,d)$ is a finite metric space , then every ideal of $C(X, \mathbb R)$ is generated by an idempotent ? [closed]

If $(X,d)$ is a finite metric space , then is it true that every ideal of $C(X, \mathbb R)$ is generated by an idempotent ?
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An ideal $I$ in a commutative ring is an intersection of prime ideals if and only if $a^2 \in I \implies a \in I $ ? [closed]

Is it true that an ideal $I$ in a commutative ring is an intersection of prime ideals if and only if $a^2 \in I \implies a \in I $ ?
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25 views

Finding powers of prime ideals from its generators and understanding generator notation

I am trying to understand ideal notation with pointed brackets and how to use it. For instance, if I had an ideal $\mathfrak{a}=\left<2,1+\sqrt{-5}\right>$, where $2$ and $1+\sqrt{-5}$ are its ...
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25 views

Norm of an ideal is finite

I want to show that the norm $N_{K/\mathbb Q}(\mathfrak{a})$ of $\mathfrak{a}$ a nonzero integral ideal of a number field $K$ is finite, and so $N_{K/\mathbb Q}(\mathfrak{ab})=N_{K/\mathbb ...
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Sum of ideals in polynomial rings

Let $ I = \lbrace g(X) \in \mathbb{Z} \;| \; g(0)\in 5\mathbb{Z}\rbrace$ Show that $I$ is an ideal in $\mathbb{Z}[X]$, and that $I = \langle 5\rangle + \langle X\rangle $. From previous parts of ...
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A question concerning to show that $V(I)$ is open if $I$ is radical ideal

Let $I:=(f_1 ,...,f_k)$ be a finitely generated ideal of $C(X,\mathbb R)$ such that $\mathrm{rad}(I)=I$, $f:=\sqrt{\sum_{m=1}^k |f_m|}=\sum _{m=1}^k g_mf_m$ where $g_i $'s are real valued ...
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Polynomial ideals

I got stuck with an exercise while preparing for my exam, and could use a hint or two to move on... Let $f(X) = a_n X^n+a_{n-1} X^{n-1}+ \cdots +a_0 \in \mathbb{Z}[X]$ with $a_0\neq 0$ Assuming that ...
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An essential right ideal in a ring

Let $S⊆R$ be rings with unity such that $S_S$ is essential in $R_S$. If $r∈R$ is a nonzero element there exists an $s_0∈S$ with $rs_0$ a nonzero element of $S$. Now, could we find a right ideal $I$ ...
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58 views

$I^{j}/I^{j+1} \cong R/I$ for any ideal I in ring R.

Let $R$ be a commutative ring with $1$ and $I$ be an ideal in it. Let $\overline{\alpha} \in I^j\setminus I^{j+1}$ and define $\theta\colon R \to I^j/I^{j+1}$ by $\theta(x)=\overline{\alpha x}$. My ...
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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 $m≥3$. 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})∈M_m(K)|\sum\limits_{i = ...
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Is this homomorphism in general surjective?

Let $R$ be a commutative ring and $I$ an ideal of $R$. Pick a fix $0 \not= a \in I$ and consider the map $\phi: R \to I$ given by $r \mapsto ra$. Is this map surjective?
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Grobner basis and subsets

Let $A$ be a subset and $I$ an ideal of polynomial ring $R=k[x_1,x_2,...,x_n]$. Is there any algorithm for deciding when $A\subseteq I$?
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Pull back image of maximal ideal under surjective ring homomorphism is maximal

Let $f :R \to S$ be a surjective ring homomorphism , $M$ be a maximal ideal of $S$ , I am writing a proof showing $f^{-1}(M)$ is a maximal ideal of $R$ , Please verify whether it is correct or not . ...
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Trying to prove $Z(I(A))=\bar A$

For every ideal $I$ of $C[0,1]$ , define $Z(I):=\{x \in [0,1] :f(x)=0 , \forall f \in I\}$ and for every $A \subseteq [0,1]$ , let $I(A):=\{f \in C[0,1] : f(x)=0 , \forall x \in A\}$ . Then ...
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90 views

Analysis of the ideals of $C[0,1]$

For every ideal $I$ of $C[0,1]$ , define $Z(I):=\{x \in [0,1] :f(x)=0 , \forall f \in I\}$ and for every $A \subseteq [0,1]$ , let $I(A):=\{f \in C[0,1] : f(x)=0 , \forall x \in A\}$ . Then ...
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prove that the quotient ring S3/T3 is isomorphic to D3

Could you please help with this question? I've already shown that T_3 is an ideal of S_3. Thanks,
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Find the splitting fields of the polynomials

What are the splitting fields for the polynomials $f(x)=x^4 + 5x^2 +4 $ and $f(x)=x^4 - x^2 - 2 $ I know that any polynomial has a splitting field and by using the proof of this fact $f(x)$ of ...
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On the Krull Dimension of Quotient Rings

Let $I\subset J \subset R$ be ideals of $R$. How can we show that $\dim(R/J) \leq \dim(R/I)$? So far I've shown that the heights meet $ht(I)\leq ht(J)$, which is fairly straight forward, but I am ...
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Is a square of a prime ideal in a UFD always primary?

More concretely, Let $R$ be a UFD and $\mathfrak{p}$ a prime ideal ideal of $R$. Does it always hold that $\mathfrak{p}^2$ is a primary ideal? I know that it always holds if $\mathfrak{p}$ is a ...
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Quotient ring of complex polynomials and ideal domain

Let f(X) = X^2 − 2X + 5 ∈ C[X] and the ideal generated by f(X) be I = f(X)C[X]. (where C(X) is the set of complex polynomials) Prove that the quotient ring C[X]/I is not an integral domain. Since ...
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How to characterize the maximal ideals of $\mathbb R[x]$ and $\mathbb C[x]$ ?

It is known that if $M$ is a maximal ideal of $C[0,1]$ then for some $r \in [0,1]$ , $M=\{f \in [0,1] : f(r)=0\}$ , can we also characterize the maximal ideals of $\mathbb R[x]$ ? I think I somewhere ...
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Prove that $I$ is a left ideal of $R$

Let $R$ be a ring. Let $z$ be a fixed element of $R$ and $J$ be an ideal of $R$. Define $I = \{r\in R|rz \in J\}$. Prove that $I$ is a left ideal of $R$. My Attempt $I \neq \emptyset$, since ...
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Maximal ideals of $\mathbb R[x] / \langle (x-a)(x-b) \rangle$ , where $a,b$ are reals and of $\mathbb R[x] / \langle x-(a+\bar a)x+a\bar a \rangle$?

How to determine maximal ideals of $\mathbb R[x] / \langle (x-a)(x-b) \rangle$ , where $a,b$ are reals ? I know it has only four ideals , the ring itself cannot be maximal . Also the zero ideal i.e. ...
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Badly behaved, but easy-to-manipulate examples of rings to test hypotheses on

In calculating examples in mathematics it's often useful to have a quite misbehaving but easy-to-manipulate object to test hypotheses on. Examples are the function $ f(x)=\begin{cases} 0 & \text{ ...
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Example to show that multiplication by ideals and intersection of submodules do not commute

The key point of question about typical proof of Krull Intersection Theorem is that multiplication by ideals and intersection of submodules do not commute. Can anyone give me an example of this? ...
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Intersection of submodules

I have question regarding intersection of submodules. Could anyone give example of a commutative ring $R$ with identity and an $R$-module $M$ such that $$IM\cap JM\nsubseteq (I\cap J)M$$ for some ...
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What are the ideals of $End(R)$?

Let $R$ be a ring (with unity if necessary) , then $End(R)$ i.e. the set of endomorphism of the ring $R$ (the set of all ring homomorphisms from $R$ to $R$ ) forms a ring under point-wise function ...
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A more structural proof using homomorphisms and similar tools that every ideal of $M_n(R)$ is of the form $M_n(I)$

Let $R$ be a ring with unity , we know that if $J$ is an ideal of $M_n(R)$ then for some ideal $I$ of $R$ , $J=M_n(I)$ . The proof I know is very tedious and uses laborious manipulations using ...
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how to identify the ideals of a ring by using canonical homomorphism?

Assume we have a quotient ring $R'=\mathbb{C}[t]/(t-1) $. How can I find the ideals of $ R' $ by using the cannonical homomorphism $ H$ from $\mathbb{C}[t] $ to $ R' $. This is my homework actually ...
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Can we show , without considering the real numbers , that $\mathcal N$ is a maximal ideal of $\mathcal C$?

Let $\mathcal C :=\{(r_n)\subseteq \mathbb Q : \forall k \in \mathbb Q^+ , \exists N_k \in \mathbb N : |r_n-r_m| < \dfrac1{k} , \forall n,m \ge N_k \}$ and $\mathcal N:=\{(r_n)\subseteq \mathbb Q ...
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Isotypic components are just simple two-sided ideals

I'm trying to show that when we decompose a semisimple ring $R$ into isotypic components $$ R \overset{_R\mathsf{Mod}}{\cong}\bigoplus_{j=1}^{k_1}{I^{(1)}_j} \bigoplus \dotsb \bigoplus \left( ...
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Extension of prime ideals to polynomial rings

For a commutative ring $R$ prove that the ideal $P[X]$ is prime if $P$ is prime ideal in $R$. I know that $$R[X]/P[X]≅(R/P)[X].$$ Also an ideal $I$ of a commutative ring $R$ is prime if and ...
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30 views

Does canonical projection of commutative rings(R to R/I) always send prime ideals to prime ideals?

R is a commutative ring with 1, I is an ideal of R. Consider the canonical projection f: R to R/I. Suppose p is a prime ideal of R then is f(p) always prime? I think if ab+I$\in$f(P) with ab$\in$ p, ...
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One-sided nilpotent ideal not in the Jacobson radical?

Problem XVII.5a of Lang's Algebra, revised 3rd edition, is: Suppose $N$ is a two-sided nilpotent ideal of a ring $R$. Show that $N$ is contained in the Jacobson radical $J: = \{ \cap\, I: I ...