Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [ideals]

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, ideals in Lie algebras.

An ideal $I$ in a ring $(R,+,\cdot)$ a subset $I\subseteq R$ such that $(I,+)$ is a subgroup of the additive group $(R,+)$ and $r\cdot x,x\cdot r\in I$ whenever $r\in R$ and $x\in I$ (i.e., $I$ is closed under multiplication by arbitrary elements).

This is the most frequent use of the name ideal, but it is used in other areas of mathematics too:

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Prime ideals of Z

I must be going crazy... We know that for an integral domain, $R, a \in R$ is prime if and only if $(a)$ is a prime ideal. So taking $R$ to be the integers and $a=2$. Obviously 2 is prime and looking at $(2) = 2\mathbb{Z} = \{0, \pm 2, \pm 4, \pm…
Tal Bashan
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prime ideal that is not max ideal

It's easy to see that $\mathbb{Z}$ is noetherian but not artinian. In my course notes there's proven: $A$ a commutative ring: $A$ artininian $\Leftrightarrow$ $A$ noetherian and Spec($A$) = Max($A$) with Spec($A$) = collection of prime ideals, so…
Koen
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Proving prime ideal

Prove that $P$ is prime if and only if it has this property: Whenever $A$ and $B$ are ideals in $R$ such that $AB \subseteq P$, then $A \subseteq P$ or $B \subseteq P$, where $P$ be an ideal in a commutative ring $R$ with $P \ne R$.
WoShiDaYe
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Question about proof of proposition $11$ in chapter $15.2$ of Dummit & Foote (3rd ed.).

Here is the relevant passage: Now, I am unsure what they mean by "passing to $R/I, \ldots$". They don´t seem to use that fact later? I mean, $R/I = R$ given that $I = 0$. So this seems entirely vacuous to me. Am I missing something? Here is…
Ben123
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Exercise with ideals (that I tried to solve involving some numbers theory)

I=<(20 + 24$\sqrt{(-7)}$, n> is an ideal. I am trying to find an n$\in$$Z$, so that I$\neq$$Z$$[\sqrt-7$]. So far my idea was to try to see how $\frac{Z}{(20 + 24\sqrt{(-7)})}$ would look like. With $a \in Z$, $\frac{a}{(20 +…
user1122379
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Isomorphism between fractional ideals

Let the restricted direct product $$V_N = (0, \infty) \times \prod_{p \mid N} \{x \in \mathbb Z_p \ : \ x \equiv 1 \mod N\} \times \prod_{p \nmid N} \mathbb Q_p^\times$$ Let $I_N$ be the group al fractional ideals of $\mathbb Q$ prime to $N$. Let…
Wolker
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Factorization of (54) in $\mathcal O_{-53}$

In $\mathcal O_{-53}$, we have $54 = 3^3 \cdot 2 = (1+\sqrt{-53})(1-\sqrt{-53})$. What's the factorization of the ideal $(54)$ in $\mathcal O_{-53}$ ?
StMan
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Ideal $I=(x^3,x^5)$ in $\mathbb Q[x]$ . It is true that $I=(x^3)$ ?What about $I=(x^5)$?

Ideal $~I=(x^3,x^5)~$ in $~\mathbb Q[x]~$ . It is true that $~I=(x^3)~$ ?What about $~I=(x^5)~$ ? My work : I was thinking to solve this with something like: $~(x^3)(1+x^2)=x^3+x^5~$ and $~(1+x^2)\in \mathbb Q[x]~$ and for $~I=(x^5)\implies…
ROSole
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Is there any particular study on the following group

The set of all principal fractional ideals of $Q(i)$ of the form $(\frac{a+bi}{c})$ where $a^2+b^2=c^2$ with multiplication of ideals form a group.
unknownMe
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How to prove that $\langle 3+8\sqrt{2},7\rangle = \langle 3+\sqrt{2}\rangle$ in the ring $\mathbb{Z}[\sqrt{2}]$?

How to prove that the ideals $\langle 3+8\sqrt{2},7\rangle$ and $\langle 3+\sqrt{2}\rangle$ are equal in the ring $\mathbb{Z}[\sqrt{2}]$? I tried using the factors and reducing to the form of the other and vice versa, but it did not work. If there…
pigeon
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Prime and maximal ideal

If I have to show that ideal A is not maximal, is it enough to show that A is not prime because it is usually easier? Every maximal ideal is prime so if we have ideal that is not prime, it can not be maximal, if I am thinking right.
Simple
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If $a$ in $R$ is prime, then $(a+P)$ is prime in $R/P$.

Let $R$ be a UFD and $P$ a prime ideal. Here we are defining a UFD with primes and not irreducibles. Is the following true and what is the justification? If $a$ in $R$ is prime, then $(a+P)$ is prime in $R/P$.
Nikolaj-K
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Morita equivalence between right and left ideals of a ring

I would like to know whether Morita equivalence is a useful tool when dealing with right and left ideals of a ring. If so, could someone illustrate it on the example of $2\times 2$ matrices? Thanks
Javier Arias
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Origin of ideals in order theory

I am trying to clarify in my head the different meanings of "ideals" in mathematics. We have ideals in Number Theory, as in Dedekind (derived from 'Ideal Complex Numbers' in Kummer), in Abstract Algebra (Ring Theory), as in Dedekind and, mostly, in…
Javier Arias
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Let $A$ be a ring. Let $I$, $J$ be two ideals of $A$.The following properties are ture.

Let $A$ be a ring. Let $I$, $J$ be two ideals of $A$.The following properties are ture. (a) The radical $\sqrt[]{\mathstrut I}$ equals the intersection of the ideals $\rho$ $\in$ V(I). (b) We have $V(I)$ $\supseteq$ $V(J)$ if and only if $J$ …
user190302
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