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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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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32 views

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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32 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 ...
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Intersection of ring and prime ideal

Give an example of an extension $B/A$ of rings, with $B$ an integral domain and a nonzero prime ideal $\mathfrak{p}$ of B such that $\mathfrak{p} \cap A=(0).$ I don't know where to begin with this.. ...
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Example of a maximal idea

Let $A$ be the set of bounded continuous functions from the set of real numbers to itself. Then $A$ is a ring under pointwise addition and multiplication. The set $I$ of all functions $f \in A$ ...
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What is $I(\{(0,0)\})$?

I have two (algebraic) sets: $X_1 = Z(x) \subseteq \mathbb{A}^2$, ie, $X_1 = \{(0,y):y \in \mathbb{K}\} \subseteq \mathbb{A}^2$ $X_2 = Z(x+y^2) \subseteq \mathbb{A}^2$, ie, $X_2 = \{(-y^2,y):y \in ...
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43 views

Prove $I \subseteq I+J$ and $J \subseteq I+J$

Let $R$ be a ring and $I$ and $J$ be the ideals of $R$. Prove Prove $I \subseteq I+J$ and $J \subseteq I+J$ I know this is very trivial, but I still need to check what I am doing is correct or not... ...
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Prove/disprove: $I \cup J$ is (always) an Ideal of $R$.

Let $I$ and $J$ be the ideals of $R$. Prove/disprove: $I \cup J$ is (always) an Ideal of $R$. Rough Sketch: Since, $I$ and $J$ are the ideals of $R$, we have $0_R \in I$ or $0_R \in J$. Hence, $0_R ...
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41 views

Multiplication and Addition tables the following:

What would be the addition and multiplication tables of $Z_2[x]/\langle x^2 + x\rangle$? I know how to do the addition and multiplication tables for normal modular arithmetic, butam not sure about ...
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What is the dimension of $\mathbb R[x] / \langle x^3-x\rangle$ as a vector space over $\mathbb R$ ?

What is the dimension of $\mathbb R[x] / \langle x^3-x\rangle$ as a vector space over $\mathbb R$ ? Can someone please give some links , articles where I can study about polynomila rings and its ...
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Definition: What is a two-sided Lie ideal of a Lie algebra?

Let $\mathfrak{g}$ be a Lie algebra and let $\mathfrak{h}$ be a subalgebra. According to wikipedia, $\mathfrak{h}$ is called an ideal of $\mathfrak{g}$ if it satisfies the condition that ...
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Variety of an ideal

Let $F$ be a field, and $\underline{x}$ be the $n$-tuple $(x_1, x_2, ... , x_n) \in F^n$. Also, denote $F[x_1, x_2, ... , x_n]$ by $F[\underline{x}]$. Let $J$ be an ideal of ...
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If $I$ is a proper ideal of $C[0,1]$ , then should there exist $a \in [0,1]$ such that $f(a)=0 , \forall f \in I$?

Let $C[0,1]$ be the ring of all real valued continuous functions under point-wise addition and multiplication . We know that for every $a \in [0,1]$ , $\{f \in C[0,1] : f(a)=0\}$ is a proper ideal of ...
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120 views

Finding a finite generating set of an ideal of monomials

My problem involves considering the ideal $I = \{ X^mY^n \mid m,n\in \mathbb{N}, m^2n>5 \}$ of $\mathbb{Q}[X, Y]$. I am asked to write down a finite generating set of $I$ and explain how I ...
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Want to show that $g\in I$ where $I$ is an ideal, given the following conditions

Let $R=K[x_1,...,x_n]$ and $I$ be an ideal of $R$, $K$ being a field Given $h\in I$, $g\in \sqrt{I}$ and $f\in\sqrt{I}$ Where $in_<(f)=in_<(h)$ and $g=f-h$. So $in_<(g) < ...
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Relation between Variety of $(I\cap J)$ and Variety of $(I)$ $\cap$ Variety of $(J)$

I was wondering whether a relationship exists between $V(I\cap J)$ and $V(I)\cap V(J)$. Where $I$ and $J$ are ideals of the ring $R=K[x_1,...,x_n]$.
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Prove the Ring Homomorphism is Surjective

Prove the following homomorphism is surjective. $$f : Z → \frac{Z}{3Z} × \frac{Z}{5Z}$$ I completely get the questions and i can prove it by working out a corresponding pre-image for all of ...
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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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53 views

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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121 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, ...
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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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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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59 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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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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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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60 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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26 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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46 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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21 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 ...