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, ...

learn more… | top users | synonyms

0
votes
1answer
85 views

Maximal ideal of polynomial ring over a subfield

Let $L/K$ be an algebraic extension of fields. Let $B = L[X,Y]$ and $A = K[X,Y]$. Suppose $a$, $b \in L$ and $m = (X-a,Y-b)$ is an ideal of $B$. Show that $m$ and $m \cap A$ are maximal ideals of $B$ ...
-1
votes
1answer
66 views

$M=I\times J $ for some $I,J$ [duplicate]

Let $R,S$ be two rings with identity. Prove that every ideal of $R\times S$ is of the form $I \times J$ where $I$ is an ideal of $R$ and $J$ is an ideal of $S$ . Obviously $I \times J$ is an ideal of ...
1
vote
2answers
73 views

In a regular local ring every ideal has a FFR

Let $A$ be a regular local ring. Then every ideal has a finite free resolution. My thoughts: it's easy to prove that every ideal $I$ has a free resolution. In fact $I$ is finite and there is a ...
3
votes
1answer
77 views

How to show that an ideal is maximal

How do you show that $\langle y^2+2, x-1 \rangle$ is a maximal ideal in $\Bbb Q[x,y]$? I know that if you add another element that is not in this ideal, you should get the whole ring, thus showing ...
2
votes
0answers
53 views

Proof review: Every maximal ideal of ring of continuous functions has the same form

Let $R$ be the ring of real-valued continuous functions on $[0,1]$. If $M$ is a maximal ideal of $R$ prove $\exists \lambda \in [0,1]$ s.t. $M = \{f(x) \in R : f(\lambda) = 0 \}$. (from Herstein ...
4
votes
3answers
50 views

Checking whether $x^2-5$ is prime but not maximal

I want to find an example of a prime ideal that is not maximal. I thought about $x^2-5$. We know that $Z[\sqrt{-5}]\cong Z[x]/(x^2-5)$ is an integral domain, therefore is $x^2-5$ prime. However, I ...
0
votes
1answer
22 views

A question on $R$-modules

Let $M$ be a non-trivial irreducible (simple) $R$-module . Let $0 \ne m \in M$ and $A(m_0):=\{x \in R: xm_0=0\}$ , then is $A(m_0)$ a maximal left-ideal of $R$ and as $R$-modules , $M$ and $R/A(m_0)$ ...
2
votes
1answer
94 views

About Relatively Prime Ideals

I am a physics Master student who has been studying abstract algebra by himself. I have two questions about relatively prime ideals. My first question goes as follows: Let $R$ be a ring, and $...
1
vote
1answer
28 views

Prove that $I^2$ is principal.

Consider the ideal $I=(2,\sqrt{-10})$ of $\mathbb{Z}[\sqrt{-10}]$. Prove that $I^2$ is principal. My Try: $I^2=(4,-10,2\sqrt{-10})$. I tried to prove that $I^2=(\sqrt{-10})$. But failed. Is my ...
2
votes
1answer
70 views

Can MAGMA write Groebner basis elements in terms of the original generators?

Consider the free algebra $F = \mathbb Q(a)\langle x, y, z\rangle$ and the ideal $$I = \langle xy - ayx, yz - zy, xz - zx - y\rangle$$ According to the following code $y^2 \in I$. ...
0
votes
0answers
34 views

Confusion regarding a statement in Atiyah-Macdonald

Atiyah-Macdonald says the following: If the ideals $a_i, a_j $ are co prime, then $\Pi a_i= \cap a_i $ What does this even mean? For example, we know that $(2), (3) $ are co prime in the ring $\...
2
votes
2answers
89 views

For $I,J$ ideals $P$ Prime ideal, show that $IJ\subset P \iff I\cap J \subset P$

Question : Prove the following equivalence $IJ\subset P \iff I\cap J \subset P \iff$ $I$ or $J \subset P$ I was able to do this $IJ \subset I$ and $IJ \subset J$ so $IJ \subset P$ $IJ \subset I$ ...
2
votes
1answer
36 views

Expressing polynomial as linear combinaion

I found these questions in Adams Introduction to Groebner bases Let $f=x^6-1$ and $g=x^4+2x^3+2x^2-2x-3$. Let $I=\langle f,g\rangle$. Calculate the polynomial that generates $I$ alone. After a ...
3
votes
1answer
101 views

Riemann-Roch Theorem and Ideals of a Ring

I found in some Math book a comment stating that the study of Ideals in ring theory à la Dedekind (all kinds of ideals? only one-sided ideals?) could be transferred to other areas (specifically, ...
0
votes
1answer
54 views

Some doubts about right ideals of a ring

I would like to know whether the following paragraph regarding right ideals and modules is correct. Any comment or help is welcome: A right ideal of $R$ is just a submodule of the right $R$-module $...
10
votes
0answers
231 views

Converse of Chinese Remainder Theorem

Chinese Remainder Theorem for commutative rings with identity Let $R$ be a commutative ring with identity. If $I, J$ are ideals of $R$ satisfying $I+J=R$, then there is an isomorphism of rings: $$R/(...
0
votes
1answer
67 views

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 ...
1
vote
1answer
88 views

Is it true that every prime ideal of height one is principal? [closed]

Is it true that every prime ideal of height one is principal ? Please help
4
votes
2answers
73 views

$R$ 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.
4
votes
1answer
100 views

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? ...
2
votes
3answers
104 views

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 $ R/\...
1
vote
1answer
54 views

$I\otimes I$ 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 $m\...
4
votes
1answer
131 views

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 ...
3
votes
1answer
116 views

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 ...
1
vote
1answer
27 views

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 $S^i(R)$...
2
votes
2answers
63 views

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 = ...
0
votes
1answer
42 views

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 $1$, ...
2
votes
1answer
38 views

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 $B$ ...
3
votes
1answer
85 views

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 $R=\{a_0+a_1x+a_2y+a_3xy\;:...
5
votes
0answers
58 views

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 $\{e_\alpha\}\...
1
vote
1answer
46 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 $(a+b)x=ax+bx$....
2
votes
2answers
91 views

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 examples,...
1
vote
3answers
46 views

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 ...
0
votes
0answers
46 views

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 ...
1
vote
0answers
49 views

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 $n$...
0
votes
1answer
58 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!
1
vote
1answer
74 views

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 $R$-...
2
votes
1answer
60 views

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 ...
0
votes
1answer
21 views

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 ...
1
vote
2answers
60 views

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 don'...
4
votes
2answers
90 views

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{ ...
1
vote
1answer
115 views

The quotient of a ring by the annihilator of an ideal

Let $R$ be a commutative ring with identity, and $I$ an ideal of $R$. Is it 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 ...
2
votes
1answer
63 views

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.
1
vote
1answer
83 views

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 $\...
3
votes
1answer
56 views

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 ...
1
vote
1answer
68 views

What is the difference between submodules of $A/\mathfrak a$ as $A$-module or as $A/\mathfrak a$-module?

If $\mathfrak a$ is an ideal of unital commutative ring $A$, then we can consider $A/\mathfrak a$ as $A$ module or as $A/\mathfrak a$ module. If $A=\mathbb Z$ there is no structural difference between ...
5
votes
2answers
99 views

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 ...
1
vote
1answer
54 views

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 ...
3
votes
2answers
91 views

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: ...
1
vote
1answer
22 views

Stone representation theorem and right(or left-) one-sided ideals in a ring

Consider Marshall Stone's representation theorem: I would like to know in which specific way, if any, it is connected with the determination of right or left one-sided ideals in a ring. Simple ...