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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Proof about the difference between right and left ideals in a ring

I have tried get a version of the proof stating that a left ideals of a ring is not, in general, a right ideal, and viceversa. Is my formulation right? Comments and corrections are welcome. I have ...
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Ring homomorphism and ideal that contains the kernel

If $f:R\rightarrow S$ is a ring homomorphism and $I$ ia an ideal of $R$ such that $ker(f) \subseteq I$ then $f^{-1}(f(I))=I$ We know that $I\subseteq f^{-1}(f(I))$ but how can use that $ker(f) ...
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Is it true that every prime ideal of height one is principal? [on hold]

Is it true that every prime ideal of height one is principal ? Please help
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$R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$

$R$ a local ring, also a PID. $I,J$ ideals from $R$. Show that $I \subseteq J$ or $J \subseteq I$ My brief attempt to try use Bezout theorem at a PID. but unsuccess.. Thanks any help.
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Is radical of finitely generated ideal finitely generated?

Let $R$ be non-noetherian commutative ring with identity and $I$ be a finitely generated ideal of $R$; say $I = (a_1, \cdots, a_n)$. Question.1 Is $\sqrt I$ necessarily finitely generated? ...
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Prove that an ideal $ \mathfrak{m} $ of a commutative ring $ R $ is maximal iff $ R/\mathfrak{m} $ is simple.

Could someone give me a hint on whether I’m on the right track or not? For sufficiency, I tried the following: Suppose that $ \mathfrak{m} $ is a maximal ideal. With the quotient map, we get $ ...
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$I\otimes I$ is torsion free for a principal ideal $I$ in domain $R$

Question is : Suppose $I$ is a principal ideal in a domain $R$. Prove that the $R$ module $I\otimes_R I$ is torsion free. Suppose we have $r(m\otimes n)=0$.. Just for simplicity assume that ...
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How does one find the Zariski closure of a set?

I've started to learn algebraic geometry this week (so I do not have much knowledge in the subjet) and, after reading the definition of the Zariski closure $V(I(S))$ of a set $S$, I've tried to do the ...
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On a theorem of Akizuki concerning the minimal number of generators of an ideal

I am looking for a theorem of Akizuki I was told by my professor. He said me that Akizuki showed in his paper "Zur Idealtheorie der einartigen Ringbereiche mit dem Teilerkettensatz" (1938) a result ...
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Show that in ascending Loewy series, $S^r(R)=R$

Let $R$ be an Artinian ring, $N$ its radical, and $r$ the smallest natural number such that $N^r=0$. Define an ideal $S^n(R)$ of $R$ recursively as follows: $S^1(R)=soc(R)$ Assuming ...
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Factoring ideals in algebraic number rings using Dedekind's theorem

Let $K \subset L=K(\alpha)$ be a number field extension with rings of integers $\mathcal{O}_K$ and $\mathcal{O}_L=\mathcal{O}_K[\alpha]$ respectively. Let $\pi$ be a prime ideal in $O_K$, and let $F = ...
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Is the ideal $\{2m + (1 + \sqrt{-6})n:m, n\in\mathbb{Z}\}$ principal in $\mathbb{Z}[\sqrt{-6}]$?

Is the ideal $\{2m + (1 + \sqrt{-6})n:m, n\in\mathbb{Z}\}$ principal in $\mathbb{Z}[\sqrt{-6}]$? I have an exercise that asks just that. As a hint it says to prove that this ideal contains $1$, ...
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A question about 1.0.3 in Grothendieck's EGA

In (1.0.3), Grothendieck states that, given non-commutative rings $A$ and $B$, a homomorphism $\varphi : A \to B$, and a left ideal $\mathfrak{J}$ of $A$, the left ideal $B\mathfrak{J}$ of $B$ ...
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Finding all ideals in a finite ring

Let $\mathbb F_2$ be the field of two elements. Consider the factor ring $$R=\mathbb F_2[x, y]/\langle x^2, y^2\rangle.$$ I want to find all ideals of $R$. Note that ...
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Compactum of Banach algebra

I need an example of Banach algebra $A$ and a left non-trivial closed ideal $I$ with all of following properties: There exists a bounded approximate identity in $I$ for $I$ i.e., a net ...
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25 views

Prove socle is ideal

In any ring $R$ define the socle as the sum of all minimal right ideals of $R$. Say we have two minimal ideals $A,B$. If $a\in A,b\in B$, then $a+b$ is in the socle. If $x\in R$, then ...
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Ideal of polynomials vanishing on $\{(x,y): x^2+y^2=1, x \neq 0 \}$

I'm reading the book "Introduction to algebraic geometry" by Hassett, and in Chapter 3, after introducing the concept of the ideal of polynomials vanishing on a set $S$, the author gives some ...
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Centre of matrix ring over skew field

Let $R$ be a semisimple ring. Show that $R$ is simple iff the centre of $R$ is a field. Book's solution: If $R$ is simple, it has the form $\mathfrak{M}_n(K)$ for a skew field $K$, and its ...
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Determinant of the change of basis for fractional ideal

Let $A$ be a fractional ideal of some number field extension $K:\Bbb Q$. Let $\omega_1, \dots ,\omega_n$ be a $\Bbb Z$ basis for $\mathcal O_K$ and let $\alpha_1, \dots ,\alpha_n$ be a $\Bbb Z$ basis ...
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How do I prove that primary ideals satisfy this property?

Let $R$ be a commutative ring. Let $Q$ be a primary ideal of $R$. Let $I,J$ be ideals of $R$ such that $IJ\subset Q$. How do I prove that $I\subset Q$ or $J^n\subset Q$ for some positive integer ...
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24 views

Looking for an example of an ideal contained in the union of other ideals, but not in any ideal individually

I'm looking for an example of the following scenario: $A, B, C $ are three ideals such that $C\subseteq A\cup B $ but $C\not\subseteq A $ and $C\not\subseteq B$. Any help would be great!
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If $R$ is a simple Artinian ring, then when is a finitely generated module free?

Here's an exercise from my book, which only gives a brief solution which leaves me very confused. Let $R$ be a simple Artinian ring, say $R=K_r$. Show that there is only one simple right ...
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If a set $S$ generates an ideal $I\subset F[x_1,x_2,\ldots,x_n]$, then there is a finite subset $S_0 \subseteq S$ which generates $I$

The question: If $I$ is an ideal in $F[x_1,x_2,\ldots,x_n]$ generated by a set of polynomials $S$, then there is a finite subset $S_0 \subseteq S$ which generates $I$. By the Hilbert Basis ...
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Let $\phi\in \mathbb{C}[V]$. Show that $\mathbf{V}_V(\phi)=\emptyset$ if and only if $\phi$ is invertible in $\mathbb{C}[V]$.

This is an exercises in Ideals, Varieties and Algorithms by Cox et al. Let $V\subset \mathbb{C}^n$ be a nonempty variety. Let $\phi\in \mathbb{C}[V]$. Show that $\mathbf{V}_V(\phi)=\emptyset$ if ...
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Is this another way of stating the Chinese Remainder Theorem?

Assume that $I + J = R$. Let $a,b \in R$. Find an element $u$ of $R$ satisfying $a + I = u + I$ and $b + J = u + J.$ I want to work on this, but I feel there's some issue of a missing theorem I ...
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Is a polynomial $f$ zero at $(a_1,\ldots,a_n)$ iff $f$ lies in the ideal $(X-a_1,\ldots,X-a_n)$?

This is probably a very silly question: If $R$ is an arbitrary commutative ring with unit and $f\in R[X]$ a polynomial, then for any element $a\in R$ we have $$f(a)=0 \Longleftrightarrow X-a ~\mbox{ ...
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The quotient of a ring by the annihilator of an ideal

Let $R$ be a commutative ring with identity and $I$ an ideal of $R$. It's true that we have an $R$-module isomorphism $$I\cong R/ann_RI,$$ where $ann_RI=\{x\in R:xr=0,\;for\;all\;r\in I\}$ is the ...
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Ideals and injective modules

Let $I$ be a left ideal of $R$. Assume that there exist element in $I$, which is not a zero divisor. How to prove that for every (left) injective $R$-module $Q$ we have $IQ=Q$ ? I need only hints. ...
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Every modular right ideal is contained in a modular maximal ideal

If $R$ is a ring, possibly without $1$, a right ideal $\mathfrak{a}$ of $R$ is modular if there exists $e\in R$ such that $r-er\in \mathfrak{a}$ for all $r\in R$. So $e$ is a left identity mod ...
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Looking for a terminology in ring theory (“ideal” which is not necessarily closed under addition )

I am wondering if there is a name for the subsets $S$ of a commutative ring $R$ such that for every $r\in R$ and every $s\in S$ we have $rs\in S$. Thus $S$ is ...
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Prime ideal $P$ of $\mathbb{Z}[x]$ such that $P \cap \mathbb{Z}=\{0\}$ is principal

The problem stated more precisely is this: Let $P$ be a prime ideal of $\mathbb{Z}[x]$ such that $P \cap \mathbb{Z} =\{0\}$. Show that $P$ is a principal ideal. I think there is a problem with my ...
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Example of a Non-Graded Ideal in a Graded Ring

A ring $S$ is said to be graded if there are additive subgroups $S_0, S_1, S_2, \ldots$ such that $S=\bigoplus_{k\geq 0}S_k$ and $S_iS_j\subseteq S_{i+j}$ for all $i$ and $j$. An ideal $I$ in a ...
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If $R$ is a commutative ring, the nilpotents form necessarily an ideal of $R$? [duplicate]

This is an algebra question from an exam a few years ago: Let $R$ be a ring, and let $N = \{a \in R: a^n = 0 \text{ for some } n \in \mathbb{N}, (n \text{ depends on } a) \}$. Prove or disprove: ...
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Graphs associated with rings and modules

There are several articles in the literature that deals with some interesting graphs associated with rings and modules. For example The zero-divisor graphs D. F. Anderson, P. S. Livingston, The ...
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Stone representation theorem and right(or left-) one-sided ideals in a ring

Consider Marshall Stone's representation theorem: https://en.wikipedia.org/wiki/Stone's_representation_theorem_for_Boolean_algebras I would like to know in whichspecific way, if any, it is connected ...
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Prove that $f$ is a nonzerodivisor on $R[x_1,\dots,x_r]/IR[x_1,\dots,x_r]$ for every ideal $I$ in $R$

Let $R$ be a Noetherian commutative ring with unity, and $S=R[x_1,\dots,x_r]$. Let $f\in S$ be a nonzerodivisor of $S$. Suppose that the ideal generated by the coefficients of $f$ is $R$. How to ...
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Uniqueness of the decomposition of an ideal

Let $ F $ be a non-empty subset of $ \{ 1,2,\dots,n\} $ and $ P_{F}=(\{x_{i}:i\in F\}) $. Let $ F_{1},F_{2},\dots,F_{m} $ be pair-wise distinct non-empty subsets of $ \{1,2,...,n\} $ and $$ ...
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Noether & Schmeidler- Hurwitz-Ideals

Consider the following page from Noether and Schmeidler's 1921 work: http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN266833020_0008&DMDID=DMDLOG_0008&LOGID=LOG_0008&PHYSID=PHYS_0013 ...
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Left Ideals and Two-Sided Ideals of Rings of Matrices

This is a question I posted in this topic: Pathologies in "rng". However, I decided that the question deserves its own thread, and I want to know if anybody can answer it. For ...
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Radical ideal in $\mathbb{R}[x,y,z]$

In $\mathbb{R}[x,y,z]$ is the ideal $I=\left\langle xz,yz\right\rangle$ radical? If $f \in I$ tried write $f=g.xz+h.yz+ax+by+c$ and conclude that $f^m \notin I$, if $m>0$, but I could not.
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Why is $(2,x)$ non-principal in $\Bbb Z[x]$? [duplicate]

Why is $(2,x)$ non-principal in $\Bbb Z[x]$? Apparently this is the case, I just read it on wiki, as a counter example to $\Bbb Z[x]$ being a PID. What is $x$ here? I mean $2$ can surely generate ...
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Invertibility of translating ideals from a ring to its localization

Let $A$ be a ring with some multiplicative subset $S$. Define $AS^{-1} = \{\frac as| a \in A, s \in S\}$. Let $I_A$ and $I_S$ be the sets of ideals of $A \subset A-S$ and $AS^{-1}$ respectively. ...
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Are these rings fields?

Are the following rings fields? 1) $\Bbb Q[x] /\langle x^2+1\rangle$ Since a polynomial ring taking values on any field is a E.D, and hence a P.I.D, this is a field iff the ideal is prime or ...
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Quotient ring is a ring homomorphism

Why is this a ring homomorphism? $$\phi:R\to R/I$$ where $I$ is an ideal, given by $\phi:r\mapsto r+I$. To be a ring homomorphism it needs to be a homomorphism of addition and multiplication, i.e: ...
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Prove that: $B/A \triangleleft R/A$ if $A \subseteq B ;\ \ A, B \triangleleft R $(ring)

Prove that: $B/A \triangleleft R/A$ if $A \subseteq B ;\ \ A, B \triangleleft R $(ring) : $ \triangleleft $ means ideal. I need this proof to continue on, I'm told it's not that hard, but I just ...
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In the ring of polynomials $F[t]$, every ideal is principal

Let $$\langle a \rangle=\bigcap_{ a \in I} I$$ $$ \langle a \rangle =\{ Ira + na, r\in R,n \in Z\}$$ What is unclear is why$$ I=\langle 0 \rangle$$ proves that I is a principal ideal. My definition ...
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42 views

Show that $\alpha_A^{-1}(I'+J')=\alpha_A^{-1}(I')+\alpha_A^{-1}(J')$, where $I',J'$ are ideals and $\alpha_A$ is a surjective ring homomorphism.

Let $\alpha_A: k[x_1,...,x_m]\rightarrow k[y_1,...,y_n]$ be a map defined by $\alpha_A(f)(y)=f(Ay)$ where $A$ is an $m\times n$ constant matrix. Let $I',J'$ be ideals in $k[y_1,...,y_n]$. ...
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Determine which of the following rings are fields.

Have I done it correctly? Determine which of the following rings are fields: a) $(\mathbb{Z}/2\mathbb{Z})[x]$/$\large_{(x^2+1)}$ b)$(\mathbb{Z}/3\mathbb{Z})[x]$/$\large_{(x^2+1)}$ My ...
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1answer
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Show that $\left\langle\alpha_A(I\cap J)\right\rangle \subset \left\langle\alpha_A(I)\right\rangle \cap \left\langle\alpha_A(J)\right\rangle $.

Let $\alpha_A: k[x_1,...,x_m]\rightarrow k[y_1,...,y_n]$ be a map defined by $\alpha_A(f)(y)=f(Ay)$ where $A$ is an $m\times n$ matrix. Show that $\left\langle\alpha_A(I\cap J)\right\rangle \subset ...
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Prove $a\mathbb{Z}[x]+x\mathbb{Z}[x]$ is a principal ideal on $\mathbb{Z}[x] \iff a=0$ or $a=1$ or $-1$.

I've tried several things, but I don't know how to properly show it. Prove $a\mathbb{Z}[x]+x\mathbb{Z}[x]$ is a principal ideal on $\mathbb{Z}[x] \iff a=0$, $a=1$ or $a=-1$. My try: Proof: ...