All Questions
Tagged with ideals maximal-and-prime-ideals
686
questions
4
votes
1
answer
234
views
Why are prime ideals proper?
As children we all learn this erroneous definition of a prime number: “a number $n\in \Bbb N$ is prime iff it’s only divided by one and itself”. Well that’s fine until the teacher asked us for ...
- 1,207
0
votes
0
answers
48
views
Consider in $Z[x]$ the polynomial $f(x)=x^2+x+1$. How do I determine whether $I=(2,f(x))$ is a prime ideal?
Consider in $Z[x]$ the polynomial $f(x)=x^2+x+1$. How do I determine whether $I=(2,f(x))$ is a prime ideal? I read that if a polynomial is primitive then the generated ideal is prime (Why?). So $(f(x))...
1
vote
0
answers
26
views
How to break symmetry of a polynomial ideal to simplify Groebner basis?
I have an ideal $I$ generated by a set of polynomials $\{ p_i \}$. There are some variable permutations to which the ideal is symmetric. By this I mean (apologies if there is a standard term for this) ...
- 950
0
votes
0
answers
47
views
Is $I_p=(p,x^2+1)$ a prime ideal of $\mathbb Z[x]$? What is the maximal ideals of $\mathbb Z[x]$ containing $I_p$ where $p=2,3,5$?
Let $I_p$ be the ideal of $\mathbb Z[x]$ generated by $p$ and $x^2+1$.
Problem: Is $I_p=(p,x^2+1)$ a prime ideal of $\mathbb Z[x]$? What is the maximal ideals of $\mathbb Z[x]$ containing $I_p$ where $...
- 1,152
0
votes
0
answers
15
views
not irreducible but prime in a non-domain [duplicate]
Consider $\mathbf Z/6\mathbf Z$ as a ring.
It is not an integral domain since it contains zero-divisors, such as the element $[3]$ for example. Note that $[3]$ is not irreducible ($[3]^2 = [3]$), yet $...
- 387
-1
votes
0
answers
38
views
Equivalent definition for minimal ideals for commutative rings
Background
The following post on minimal ideals is a continuation and a counterpart to the following post on maxmial ideals. The quoted materials are taken from the following sources;
Fundamentals of ...
- 3,683
2
votes
1
answer
39
views
Question about showing $(x,y)$ is a maximal ideal of $\Bbb{Q}[x,y]/F[x,y]$ [duplicate]
Background
Theorem 1: Let $M$ be an ideal in a commutative ring $R$ with identity $1_R$. Then $M$ is a maximal ideal if and only if the quotient ring $R/M$ is a field.
Exercsie 1: Prove that $(x)$ ...
- 3,683
0
votes
0
answers
35
views
Understanding Radicals as Intersections of Prime Ideals and as the Preimage of a Nilradical
I ran into a proposition that the radical $r(a)$ equals the intersection of the prime ideals containing $a$. Then it is said that the latter can be understood through the following result: Let $R$ be ...
- 557
1
vote
1
answer
45
views
A commutative ring with unity which is also reduced having exactly two minimal prime ideals.
Let $R$ be a commutative reduced ring with unity. I want to find an example of such a ring which contains exactly two minimal prime ideals. For example if we take $\mathbb{Z_6}$ then $\langle2\rangle$ ...
- 533
-1
votes
1
answer
25
views
Is inclusion of a prime ideal in a non-prime ideal possible?
Let $R$ be a ring which has $p$ as a prime ideal. Can there exist a non-prime proper ideal $m$ of $R$ such that $p$ ⊊ $m$?
0
votes
1
answer
45
views
I am getting very confused by the definition of a minimal prime ideal
I am trying to prove that in an affine scheme $\operatorname{Spec}A$ that an irreducible component can be written as the vanishing locus $V(\mathfrak p)$ for a minimal prime ideal $\mathfrak p$, but I ...
- 3,431
0
votes
0
answers
28
views
Does a non-commutative ring with unity necessarily have a maximal two-sided ideal?
Let $R$ be a non-commutative ring with unity. By Zorn's lemma, $R$ must have a maximal left ideal and a maximal right ideal.
Then, does the ring $R$ necessarily have a maximal two-sided ideal?
- 923
4
votes
2
answers
225
views
Given a ring $R$ and a maximal ideal $I$, must the set $\{J \subsetneq I\}$ have the property that every element is contained in a maximal element?
Let $R$ be a commutative ring with unity and $I \subsetneq R$ an ideal that is maximal among the proper subideals of $R$. Let $K \subsetneq I$ be an ideal. Must it necessarily be true that there ...
- 1,649
1
vote
0
answers
42
views
Long exact sequence of intersected prime ideals
I'm considering a commutative local noetherian ring $R$ (in fact, $R$ is even Cohen-Macaulay) along with a number of prime ideals $I_i$. I can then construct the exact sequence
\begin{equation}
0\...
- 41
0
votes
1
answer
40
views
On principal ideals and GCD
I am currently studying algebraic number theory and came across the following exercise:
Let $A$ and $B$ be two ideals in the ring of integers $R$ of some number field. Then there exists some $\alpha\...
- 7,193