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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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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28 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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63 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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45 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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26 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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38 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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72 views

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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58 views

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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46 views

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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45 views

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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44 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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50 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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42 views

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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48 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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29 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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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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1answer
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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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33 views

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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$I^{j}/I^{j+1} \cong R/I$ 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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1answer
21 views

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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97 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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1answer
25 views

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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1answer
38 views

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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1answer
51 views

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 ...
2
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60 views