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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True or false question about polynomial ring

Let, $\mathbb{R}[x]$ be a polynomial ring and let $J = (x)$. True/false: $J$ consists of all the polynomials of $\mathbb{R}[x]$ whose constant terms are $0$. I know $J=(x)$ is a maximal ideal of ...
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Show that the ring of integers $A$ of the cubic field $\mathbb Q[x]$ with $x^3=2$ is principal.

Show that the ring of integers $A$ of the cubic field $K=\mathbb Q[x]$ with $x^3=2$ is principal. The hint given in the book is to majorize the discriminant of $A$ by $D(1,x,x^2)$ and then use the ...
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Maximal ideal of $\mathbb{R}[x]$

Let $\mathbb{R}[x]$ be a ring and let $J = (x)$. Prove that $J$ is a maximal ideal of $\mathbb{R}[x]$
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Ideal in $\mathbb Z[x]$ which is not two-generated

I can prove that the ideal $(4, 2x, x^2)$ in $\mathbb Z[x]$ is not principal. But I failed to prove that this cannot be generated by two elements. It's really difficult for me. Would you give me a ...
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Let P be a proper left ideal of R. Want to show that if P is comaximal with every non zero 2 sided ideal of R, Core(P) = {0}.

Let P be a proper left ideal of R. Want to show that if P is co-maximal with every non zero 2 sided ideal of R, Core(P) = {0}. The definition I am using of comaximal is: "I is comaximal with J if ...
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Show that $R/(I \cap J) \cong (R/I) \times (R/J) $

My question actually follows from this one: Show that if $I + J = R$, then $R/(I \cap J) \cong R/I \times R/J$ What I don't understand is why is it necessary for $I+J=R$, in order for $$ ...
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All prime ideals are maximal - Counterexample

I would like to know of some simple counter examples to the statement "ALL prime ideals are maximal" I say counter examples because I think the statement isn't true.
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Further examples of Principal Ideal Domain that are not Euclidean Domains

In several courses of algebra, I've heard that not all PIDs are EDs, and the canonical example is $\mathbb{Z}\left[\dfrac{1+\sqrt{-19}}{2}\right]$ which I've heard over and over. Some cursory research ...
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Question about ideals

I believe the following is a true statement, but I am unsure, so I wanted to check with people. If $p$ is an irreducible polynomial in $n$ indeterminates then $(p)$, the ideal generated by it, is ...
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In every ring with unity satisfying ACC, every ideal is finitely generated; can we prove it without assuming Axiom of choice?

Assuming Zorn's lemma, we can prove that in every ring with unity satisfying Ascending-Chain-Condition, every ideal is finitely generated. Is this statement equivalent to Zorn's lemma? Can we prove it ...
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68 views

In any commutative ring with unity, every prime ideal is finitely generated implies every ideal is finitely generated; can it be prove without A.C.?

Assuming Zorn's lemma, "In any commutative ring with unity, if every prime ideal is finitely generated, then every ideal is finitely generated". Is the converse true, i.e. if in any commutative ring ...
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Show that the quotient ring R/N has no non-zero nilpotent elements. [duplicate]

An element $x$ in a ring $R$ is called nilpotent if $x^n=0$ for some $n\in \mathbb N$. Let $R$ be a commutative ring and $N=\{x\in R\mid \text{x is nilpotent}\}$. (a) Show that $N$ is an ideal in ...
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29 views

What does the power of an ideal *mean*?

I am stumped trying to understand Silverman's definition of $\operatorname{ord}_P(f)$, the (normalized) valuation on $\bar K[C]_P$ (which denotes the localization of a curve $C$'s coordinate ring at ...
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Ideals Generated by polynomials

So I am currently studying a course in commutative algebra and the main object that we are looking at are ideals generated by polynomials in n variables. But the one thing I don't understand when ...
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66 views

example of proper ideal of C[x,y]

I am stuck in this problem for a while, and the main idea will be important for some exercises, so I really want to know how to find an example like this I need an example of an proper ideal, ...
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46 views

Other definition for a local ring

Suppose $R$ a ring with 1 and let $U (R)$ denote its invertible elements. If $(M = R \setminus U(R), +)$ is a group, show that $M$ is a left ideal of $R$. I know that $U (R) M \subseteq M$. But ...
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32 views

Polynomial ring and maximal ideal

I am really stump in this problem. Prove that $(x,y)$ and $(2,x,y)$ are prime ideals in $Z[x,y]$ but only the latter ideal is a maximal ideal.
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Stuggling to understand ideal powers

In my current algebraic number theory course we have defined the multiplication of 2 ideals as the smallest ideal containing all products of elements of both, [i.e: let I and J be ideals of a ring ...
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27 views

Can we characterize those Euclidean domains $D$ for which $D/I$ is finite for any ideal $I \ne \{0\}$ of $D$?

Let $I$ be any ideal of $\mathbb Z[i]$ , then as $\mathbb Z[i]$ is euclidean domain , so $I=(z)$ for some gaussian integer $z$ ; so we can write every element of $\mathbb Z[i] / I$ as ...
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A principal ideal domain if and only if proof

Show an ideal $(p)$ in a principal ideal domain in a maximal ideal if and only if $p$ is irreducible. This is a new concept i do not know how to go about this.
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Maximal ideals and Prime ideals.

Ok, I am new to the concepts of maximal ideals and prime ideals. I know the definitions for both, but I am kind of stuck with understanding the examples. So, any help would be much appreciated. ...
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42 views

Decomposition of a maximal ideal as a union of smaller prime ideals

Let $K$ be a field, $S=K[X,Y]$ the polynomial ring in two variables and consider the ideal $M=\langle X,Y\rangle$ (ideal generated by $X$ and $Y$). Show that $M$ is a union of strictly smaller prime ...
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How to show that the ideal $(X^{3},XY,Y^{n})$ of $K[X,Y]$ is primary?

I'm working on a problem in Sharp's Steps in commutative algebra, to be precise exercise 4.28 which states the following: Let $K$ be a field and $R = K[X,Y]$ be the polynomial ring in the ...
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Annihilator of a maximal ideal in a ring

Let R be a ring and M a maximal ideal in R. Prove or disprove: If M is contained in the set of zero divisors of R, then ann(M) is not 0. It is easy to see that the statement is true when M is ...
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Do we really need Zorn's lemma to prove the existence of prime ideals?

Let $A$ be a ring, $S$ a multiplicative set, and $I$ an ideal of $A$ disjoint from $S$; then there exists a prime ideal $P$ of $A$ containing $I$ and disjoint from $S$. The author of the book ...
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Semisimple implies complete reducibility

Why does a semisimple Lie algebra imply complete reducibility? I have that a semisimple Lie algebra is a Lie algebra with no non-zero solvable ideals. Complete reducibility means that every invariant ...
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On units in subrings ( or ideal ) and quotient ring of ring with unity

Let $R$ be a finite ring with unity and $S$ be an ideal ( or subring ) , let $R^*$ be the group of units of $R$ and $S^*:=R^* \cap S$ , then does $|S^*|$ divide $R^*$ ? Moreover , if $S$ is an ideal ...
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Why in a graded ring $A$ finitely generated that's an algebra over a field $K$ every maximal ideal is a $K$-subspace?

Probably this question has already been asked, but I'm very bad in find old question and I searched for half an hour, so I'm asking it again. I suppose that's true beacuse my professor used this ...
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Show any straight line is irreducible

Show that any straight line in $\mathbb{F}^{n}$ is irreducible, where F is an infinite field. I know V($ax+b$) would be a variety that represents any straight line and then V is irreducible if I(V) ...
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Is a prime principal ideal which is not maximal among principal ideals always idempotent?

Let $R$ be a commutative ring with identity, $P$ a prime principal ideal of $R$. Suppose that there exists a proper principal ideal $I$ of $R$ which is strictly larger than $P$ (i.e. $R\supsetneq ...
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absolutely convex ideals and $F$-spaces.

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 $$\mathcal{O}_K\cong \mathbb Z\left[\frac{1+\sqrt d}{2}\right] \cong\frac{\mathbb Z[x]}{\left( x^2-x-\frac{d-1}{4} \right)}.$$ This ...
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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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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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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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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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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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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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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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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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35 views

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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$I^{j}/I^{j+1} \cong R/I$ 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 ...