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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For ideals $I=(8)$ and $J=(5+5i)$ in $\mathbb{Z}[i]$, what are $IJ$, $I+J$ and $I\cap J$?

Let $I=(8)$ and $J=(5+5i)$ be ideals in $\mathbb{Z}[i]$. How do I find $x,y,z\in\mathbb{Z}[i]$ such that: $IJ=(x)$, $I+J=(y)$, $I\cap J=(z)$? Is it correct that $y=13+5i$ and $x=40+40i$?
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Prove that $\operatorname {ht}(p/a)\leq \operatorname {ht}(p)\leq \operatorname {ht}(p/a)+n$

In the Milne's book A Primer of Commutative Algebra, pg. 100, there's a proof that $\operatorname {ht}(p/a)\leq \operatorname {ht}(p)\leq \operatorname {ht}(p/a)+n$. I understand the first inequality, ...
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$A_i \subset$ ring $R$ st $\Delta A_i \equiv \{ b-c: b,c \in A_i\} = $ an ideal of $R$, then $\bigcap_i (\Delta A_i) = \Delta (\bigcap_i A_i)$.

Let $A$ be any subset of a ring $R$. $A$ is called a delta generating set of an ideal if $\Delta A = \{b - c: b,c \in A\}$ forms an ideal of $R$. Let $A_i$ be any collection of delta generating sets ...
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What does the ideal $I$ of $\mathbb{Z}[X]$ generated by $x$ and $2$ look like? [duplicate]

What does the ideal $I$ of $\mathbb{Z}[X]$ generated by $x$ and $2$ look like? I don't know how to put it into terms of more explicit set notation
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What would an ideal I generated by e.g. 15 and 12 look like? What would the quotient ring $\mathbb{Z} / I$ look like?

What would an ideal I generated by e.g. 15 and 12 look like? What would the quotient ring $\mathbb{Z} / I$ look like? How do I find a formal representation for this quotient ring? Thanks
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41 views

Given a ring $R$, $R=(1)$ is a principal ideal

An ideal generated by the element $a$ is defined to be the intersection of all ideals containing $a$. My book says $R=(1)$ is a principal ideal, and I know how to convince myself of this using ...
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Correspondence between nilpotents and between idempotents

It is well-known and easily proved that whenever $R$ is a commutative ring with unity and $S$ is a multiplicative subset of $R$, each ideal of the localization ring $R_S$ is an extended ideal (with ...
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Prove that $m_1m_2\ldots m_r=n_1n_2\ldots n_s$ implies $r=s$ for distinct maximal ideals

Let $R$ be a commutative ring where $m_1,m_2,\ldots,m_r$ and $n_1,n_2,\ldots,n_s$ are maximal ideals such that $m_1m_2\ldots m_r=n_1n_2\ldots n_s$ and $m_i \neq m_j$, $n_i \neq n_j$ if $i \neq j$. ...
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Proving $R/J$ is local, where $R=k[\Gamma]$ and $J=(x^1)\unlhd R$.

Let $\Gamma$ be the set of symbols of the type $x^q$, where $q\in\Bbb Q, \;q\ge0$. Setting $x^{q_1}\cdot x^{q_2}:=x^{q_1+q_2}$, $(\Gamma,\cdot)$ becomes a semigroup. Let then $k$ be a field. Let's ...
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On publication regarding right ideals of a ring and the sublanguages of science [closed]

As some of you may know (or may experience by searching some of my threads), I have been working on the applications of right ideals of a ring to the study of language (in particular, to the so-called ...
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If $\phi:A\to B$ is a ring homomorphism, why does there exist $\psi:\text{spec}(A)\to \text{spec}(B)$?

Let $\phi:A\to B$ be a ring homomorphism, where $A$ and $B$ are commutative rings. We know that if $q$ is a prime ideal in $B$, then $\phi^{-1}(q)$ is a prime ideal in $A$. Hence, there exists a ...
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Any ideal is an extended one

It is true for any commutative rings $S$ and $T$ with $1$ and any ring homomorphism $f:S\to T$ that the set $E$ of extended ideals in $T$ equals $\{J\mid J^{ce}=J\}$. In fact, if an ideal $J$ of $T$ ...
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Factorization of ideal in field $\mathbb{Q}(\sqrt[3]{2})$ and its normal closure

So far I've worked only with quadratic fields, and I'm not sure how to work with 3rd roots. I have ideal $(5)$ and need to factor it in $\mathbb{Q}(\sqrt[3]{2})$ and its normal closure. I know that ...
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Extension of idempotent ideals

Let $R$ be a Noetherian commutative ring with $1$. If $R[[x]]$ denotes the ring of formal power series over $R$ and $I$ is an idempotent ideal of $R$ I want to know whether the extension of $I$ in ...
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Let $(R,M)$ be a local ring. Suppose that $R$ is noetherian and let $I,J \unlhd R$ such that $J \subseteq I$. Prove that the following are equivalent.

Let $R$ be a local ring with maximal ideal $M$. Suppose that $R$ is noetherian and let $I,J$ be ideals of $R$ such that $J \subseteq I$. Consider the following statements: 1) Every minimal set of ...
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29 views

Which of the sets are ideals and maximal ideals?

The exercise asks me to prove which of the sets are ideals, and if they are, which of those are maximal. I have these 4 cases: $$ a) J = \{f(x)\in \mathbb{Q}[x]: f(1)=f(7)=0 \} \\b) J = \{f(x)\in ...
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Proving $\mathbb{Q}[\sqrt{2}] = \{f(\sqrt{2}): f(x) \in \mathbb{Q}[x]\} = \{x+y\sqrt{2}:x,y\in\mathbb{Q}\}$

I need to prove that: $$\mathbb{Q}[\sqrt{2}] = \{f(\sqrt{2}): f(x) \in \mathbb{Q}[x]\} = \{x+y\sqrt{2}:x,y\in\mathbb{Q}\}$$ Well, $ \{f(\sqrt{2}): f(x) \in \mathbb{Q}[x]\} $ is the set of ...
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Proving that if $p(x)$ divides $f(x)g(x)$ then $p(x)$ divides $f(x)$ or $g(x)$

I need to somehow prove that if $p(x)$ is irreducible and divides $f(x)g(x)$ then $p(x)$ divides $f(x)$ or $g(x)$. I've been given the hints that I should use the theorems: $p(x)$ is irreducible ...
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Proving that $x^3-2$ is irreducible over $\mathbb{Q}$

I need to prove that $x^3-2$ is irreducible over $\mathbb{Q}$. It is the same as proving that $\mathbb{Q}[x]\cdot(x^3-2)$ is a maximal ideal of $\mathbb{Q}[x]$. This is the same as saying that every ...
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Given $I=\langle xy, xz+z(y^2-z^2)\rangle$, prove that $I=\langle x, z(y^2-z^2)\rangle \cap \langle y, xz-z^3)\rangle $.

This is Exercise 3c. from Chapter 9, Section 7 of Ideals, Varieties, and Algorithms by Cox et al. Given $I=\langle xy, xz+z(y^2-z^2)\rangle$, prove that $I=\langle x, z(y^2-z^2)\rangle \cap ...
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84 views

Examples of non-principal free ideals

If $R$ is a commutative ring, and $I\subset R$ is a non-zero free ideal, then it is principal generated by a non-zerodivisor. If $R$ is a non-commutative ring having the IBN property and $I\subset R$ ...
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25 views

Maximal right ideal of $\mathbb{H}[x]$

Hi I'm trying to prove the right ideal $(x-i)\mathbb{H}[x]$ of $\mathbb{H}[x]$ is maximal. I've tried defining a surjective function $f:\mathbb{H}[x] \to \mathbb{H}$ by $g(x) \mapsto g(i)$ and using ...
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Intersection $I \cap \mathbb{Z}$ where $I$ is an ideal of $\mathbb{Z}[X]$

is there a reasonable algorithm that allows, given finitely many generators of an ideal $I$ of $\mathbb{Z}[X]$, to find the intersection $I \cap \mathbb{Z}$? Thank you.
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Proving that there only finitely many minimal prime ideals of any ideal in Noetherian commutative ring

Currently, I'm trying to solve a problem from a textbook: Let $R$ be a commutative Noetherian ring with identity, and let $I \subset R$ be a proper ideal of $R$. Then we know that set of prime Ideals ...
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Is the Zariski topology the same as the cofinite topology?

Let $R$ be a commutative ring, $spec(R)$ be the set of all prime ideals on $R$. For any ideal $I$ on $R$, we define the $V_I$ to be the set of all prime ideals containing $I$. We define the Zariski ...
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Idempotent ideals having only idempotents

I search for sufficient conditions for a ring $R$ so that any idempotent ideal constitutes only of idempotent elements of $R$. Of course, in the commutative case, any finitely generated idempotent ...
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How do we prove the “lying over” property for integral extensions?

Let $R \subset S$ be an integral extension of commutative rings. Then if $P \subset R$ is prime, there exists a prime ideal $Q \subset S$ such that $Q \cap R = P$. My D&F book says look at ...
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Which ideal contains which? Or are they the same?

If I'm understanding this correctly, $13$ in $\mathcal{O}_{\mathbb{Q}(\sqrt{-23})}$ is irreducible but not prime. From A & W we see that $x^2 \equiv -23 \bmod 13$ has solutions and therefore ...
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There is no $n$ such that every ideals of $K[X,Y]$ is generated by $n$ elements [closed]

How to prove that there does not exist any integer $n$ such that all ideals of $K[X,Y]$ can be generated by $n$ elements?
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Idempotent direct summands of rings

I know that if an ideal $I$ is a direct summand of a ring $R$ then it is an idempotent ideal, i.e. $I^2=I$. My question concerns the rings all of whose idempotent ideals are direct summands. ...
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Let $I$ be an ideal of ring $R$ contained in another ring $S$: what is $IS$?

In fact, I know the definition of $IS=\{$linear combinations of ab with a in I and b in S$\}$ and that it is an ideal in $S$. I was wondering what is the most general framework this fitted into? e.g. ...
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How does $R/(R \cap I)$ sit inside of $S/I$, where $R \subset S$ is an integral extension.

If $S \supset R$ is an integral extension of commutative rings, $1_S \in R$, and $I \subset S$ an ideal in $S$, then I can see that since $\exists f \in R[x]$ such that $f(s) = 0$, $\forall s \in S$, ...
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A probable equality between two ideals

Let $I$ and $J$ be ideals of a ring $R$. I want to know whether the ideal $(I+J)^2$ equals $I^2+IJ+JI+J^2$. By taking elements and using the definition of product of two ideals $I$ and $J$ as the set ...
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51 views

Norm of powers of a maximal ideal (in residually finite rings)

Let $A$ be a residually finite integral domain and $M$ a maximal ideal in $A$. Is this true that $$|A/M^k|=|A/M|^k \quad (k\in\textbf{N}) \quad ?$$ In Hirano's article On Residually Finite Rings we ...
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Prove that the ring $\mathbb{Z}[x]$ is not a PID [duplicate]

This is from P.Aluffi's book "Algebra: Chapter 0". The author gives an example of the ring that is not PID. He claims that the ideal $(2,x)$ can't be generated by a single element of $\mathbb{Z}[x]$. ...
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Show that $(T^2, T^3)$ is not a principal ideal in $\{ a + T^2 f\mid a \in \mathbb{Q}, f \in \mathbb{Q}[T] \}$.

Let $R = \{ a + T^2 f \mid a\in \mathbb{Q}, f \in \mathbb{Q}[T] \}$. Show that the ideal $(T^2, T^3)$ is not principal in $R$. So far, I've shown that the the set of invertible elements of $R$ ...
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Norm of powers of a maximal ideal

Let $A$ be a integral domain and $M$ a maximal ideal in $A$ such that the quotient $A/M$ is a finite ring (and thus a finite field). Is it true, in general, that $$|A/M^k|=|A/M|^k \quad ...
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Idempotent ideal matrices

Let $I$ be a nilpotent ideal in a ring $R$. It could be easily deduced, by the definition of product of ideals, that the full matrix ring $\mathbb M_n(I)$ for any natural number $n$ is also ...
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Proving that the canonical ring homomorphism in $\mathbb{Z}[i] / \left< 5+3i \right>$ is surjective

Let $R=\mathbb{Z}[i]$ be the ring of Gaussian integers. Let $z=5 + 3i$ and let $I=\left< z \right>$. Let $\phi: \mathbb{Z} \rightarrow R/I$ be the canonical ring homomorphism. I am trying to ...
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Idempotent subideals of $J(R)$

If $R$ is a unital ring, it is well-known that its Jacobson radical $J(R)$ contains no non-zero idempotent element of $R$. My question: Is there a ring $R$ such that $J(R)$ contains a non-zero ...
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The algebraic set $V$ is connected if and only if the coordinate ring $k[V]$ is not the direct sum of two nonzero ideals.

Let $V$ be an affine algebraic set in $\Bbb{A}^n$. Then $V$ is connected in the Zariski topology on $V$ if and only if $k[V] = k[\Bbb{A}^n]/I(V)$ is not the direct some of two ideals. I'm stuck ...
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Coprime elements in a PID satisfy that any of their powers are coprime

I have recently met this problem in my abstract algebra dealing with PID rings and coprimes stating: Let D be a PID ring $ a,b \in D $ two coprime elements. We are to show that for all $ m,n \in ...
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Proper ideal $I \implies \exists $ prime ideals $P_i$ such that $P_1 \cdots P_n \subset I$.

Let the below ideals be in a commutative Noetherian ring $R$. Corollary 22. (3) There are prime ideals $P_1, \dots, P_n$ (not necc. distinct) $\supset I$ such that $P_1\cdots P_n \subset I$. (Out ...
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Why is this sequence of ideals an ascending chain? In proof of irreducible ideals are primary.

Let $P$ be an irreducible ideal in commutative ring $R$. Suppose $ab \in P$, $a \notin P$, and define $A_n = \{b^n x : x \in R\} \cap P$. Then "clearly $A_n \subset A_{n+1}$" says D&F. But ...
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1answer
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If $Q$ is an ideal whose radical is a maximal ideal, then $Q$ is a primary ideal.

If $Q$ is an ideal whose radical is a maximal ideal, then $Q$ is a primary ideal. I wanted to prove this using these facts: $Q$ is primary if and only if every zero divisor in $R/Q$ is ...
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Shortcut to: the radical of a proper ideal $I$ is the intersection of all prime ideals $\supset I$.

From Dummit & Foote: Proof: Passing to $R/I$. Proposition 11 shows that it suffices to prove this result for $I = 0$ ... Proposition 11: ... $(\text{rad}\ I)/I$ is the nilradical of ...
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Augmentation ideal and the abelianization of $G$

On a qual problem recently, I came across the following fact: If $G$ is a finite group, and $\mathfrak{a}$ is the augmentation ideal of the integral group ring $\mathbb{Z}G$, then ...
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1answer
38 views

$R$ semisimple artinian ring $\Rightarrow\varphi(R)$ is such [duplicate]

Let $R$ be a semisimple artinian ring, i.e. a right artinian ring with no nonzero nilpotents right ideals. We know that: $R\simeq M_{n_1}(D_1)\times\dots\times M_{n_t}(D_t)$ for ...
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1answer
27 views

Height of $p/a$ is less or equal than the height of $p$.

Let $a\subset p$ be prime ideals in a noetherian ring $A$. I want to prove that height of $p/a$ is less or equal than the height of $p$. The hint to prove that is that exists a bijection between ...
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57 views

$p$ is a positive integer and $(p)$ is a maximal ideal in the ring $(\mathbb Z, +,\cdot)$, then $p$ is a prime number

I need to prove: $p$ is a positive integer and $(p)$ is a maximal ideal in the ring $(\mathbb Z, +,\cdot)$, then $p$ is a prime number. My attempt: 1) $(p)$ is a maximal ideal, so it is a prime ...