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Questions tagged [maximal-and-prime-ideals]

For questions about prime ideals and maximal ideals in rings.

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. They are defined as ideals such that $ab\in I$ implies $a\in I$ or $b\in I$. A maximal ideal ideal is an ideal which is maximal w.r.t. inclusion.

In the ring of integers maximal and prime ideals coincide. They are the sets that contain all the multiples of a given prime number, together with the zero ideal.

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Classification of prime ideals of $\mathbb{Z}[X]$

Let $\mathbb{Z}[X]$ be the ring of polynomials in one variable over $\Bbb Z$. My question: Is every prime ideal of $\mathbb{Z}[X]$ one of following types? If yes, how would you prove this? $(0)$. $(f(X))$, where $f(X)$ is an irreducible…
Makoto Kato
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A (non-artificial) example of a ring without maximal ideals

As a brief overview of the below, I am asking for: An example of a ring with no maximal ideals that is not a zero ring. A proof (or counterexample) that $R:=C_0(\mathbb{R})/C_c(\mathbb{R})$ is a ring with no maximal ideals. A homework question in…
J. Loreaux
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Examples of prime ideals that are not maximal

I would like to know of some examples of a prime ideal that is not maximal in some commutative ring with unity.
tmpys
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One-to-one correspondence of ideals in the quotient also extends to prime ideals?

I'm beginning to learn some Grothendieck's algebraic geometry and I have a doubt about a property of commutative algebra. For a comm. ring $A$ and an ideal $I$ of $A$, does the one-to-one correspondence between ideals of the quotient $A/I$ and…
Bogdan
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Proof that ideals in $C[0,1]$ are of the form $M_c$ that should not involve Zorn's Lemma

I am learning abstract algebra by myself using Dummit & Foote, and sometimes by working very hard I find very complicated non-proofs of things and I have no idea if they are the best or the worst ideas for those kind of things out there (obviously…
Patrick Da Silva
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Maximal ideals in the ring of real functions on $[0,1]$

Let $S$ be the ring of all continuous functions from $[0,1]$ to $\mathbb R$. How to prove that all maximal ideals of $S$ have the form $M_{x_0}=\{f\in S \mid f(x_0)=0\}$? Thanks in advance.
M.H
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Prove that prime ideals of a finite ring are maximal

Let $R$ be a finite commutative unitary ring. How to prove that each prime ideal of $R$ is maximal?
Alex
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Prime ideals in a finite direct product of rings

Let $S=\prod_{i=1}^{n}{R_i}$ where each $R_i$ is a commutative ring with identity. The prime ideals of $S$ are of the form $\prod_{i=1}^{n}{P_i}$ where for some $j$, $P_j$ is a prime ideal of $R_j$ and for $i\neq j$, $P_i=R_i$.
Andrew Chiriac
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Must a ring (commutative, with 1), in which every non-zero ideal is prime, be a field?

An early exercise in Irving Kaplansky's commutative rings asks: Let R be a ring. Suppose that every ideal in R (other than R) is prime. Prove that R is a field. This is easy if we assume the zero ideal is prime. But is this assumption necessary? If…
David Holden
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What's so special about a prime ideal?

An ideal is defined something like follows: Let $R$ be a ring, and $J$ an ideal in $R$. For all $a\in R$ and $b\in J$, $ab\in J$ and $ba\in J$. Now, $J$ would be considered a prime ideal if For $a,b\in R$, if $ab\in J$ then $a\in J$ or $b\in…
galois
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Spectrum of $\mathbb{Z}^\mathbb{N}$

Is anything known about the spectrum of $\mathbb{Z}^{\mathbb{N}}$? Notice that the fiber of $\mathrm{Spec}(\mathbb{Z}^{\mathbb{N}}) \to \mathrm{Spec}(\mathbb{Z})$ at a non-zero prime ideal $(p)$ is the spectrum of $\mathbb{Z}^{\mathbb{N}}/(p) \cong…
Martin Brandenburg
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Number of prime ideals of a ring

Could anyone tell me how to find the number of distinct prime ideals of the ring $$\mathbb{Q}[x]/\langle x^m-1\rangle,$$ where $m$ is a positive integer say $4$, or $5$? What result/results I need to apply to solve this problem? Thank you for your…
Marso
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How to factor in cubic extensions?

Working within the field $K=\mathbb{Q}(\sqrt[3]{n})$, for any cube root of $n$, how does one factor the unramified rational prime ideals $(p)$? For starters, I'm relatively new to this and not too sure I completely understand factoring in these…
Dave huff
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Show that any prime ideal from such a ring is maximal.

Let R be a commutative ring with an identity such that for all $r\in$ R, there exists some $n>1$ such that $r^n = r$. Show that any prime ideal is maximal. (Atiyah and MacDonald, Introduction to Commutative Algebra, Chapter 1, Exercise 7.) Any…
user41916
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Showing that the fiber of a morphism of schemes over a point is homeomorphic to the preimage

Suppose I am given a morphism of schemes $f: X \longrightarrow Y$ and suppose that for any point $y \in Y$, $\kappa(y)$ denotes the residue field. I would like to show that the fibered product $$X \times_{Y} \text{Spec}(\kappa(y))$$ is homeomorphic…
Luke
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