# Questions tagged [maximal-and-prime-ideals]

For questions about prime ideals and maximal ideals in rings.

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### Prime radical of a ring

Let $\mathbb{Z}$ be the ring of integers and $a,b,c \in \mathbb{Z}$. Consider $R= \Bigg\{\begin{pmatrix} a & c\\ 0 & b \\ \end{pmatrix} \big| a-b\equiv c\equiv 0\mod2 \Bigg\}$. ...
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### 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 ...
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### 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?
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### Is there a commutative ring with unity, zero divisors and only trivial ideals?

I know this question can't be true, otherwise we have a maximal ideal $\{0\}$ that's not prime. My textbook has the following Theorem. In a commutative ring with unity, all maximal ideals are prime. ...
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### Maximal ideals of subring of the rationals [duplicate]

Let p be a prime number, Consider the subring of $\mathbb{Q}$ $$\mathbb Z_{(p)} = \{ a/b: gcd(a, b) =1 \text{ and } p\text{ does not divide } b\}$$ I need to show that there is only one maximal ...
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### Localization using a prime ideal

I'm approaching localization of rings using multiplicatively closed sets, and the obvious case of when we take the complementary of a prime ideal of a ring, that is always multiplicatively closed. ...
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### Finding if an ideal is the radical of another one...

suppose we have, in the ring $\mathbb{Z} [x,y,z,w,v]$, the following polynomials: $f=xw-yz$, $g=x^2z-y^3$, $h=yw^2-z^3$, $k=xz^2-y^2w$. The question is to prove that $I=(f,g,h,k)$ is the radical ideal ...
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### Let $A$ be a commutative local noetherian ring and let $I$ be a proper ideal of $A$. Prove that $\bigcap_{n=1}^\infty I^n = 0$.

Let $A$ be a commutative local noetherian ring and let $I$ be a proper ideal of $A$. Prove that $\bigcap_{n=1}^\infty I^n = 0$. The first thing I tried was to see that \displaystyle\bigcap_{n = 1}^\...
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### Criterion for spanning and linear independence of the Zariski cotangent space of $R_p$ that doesn't use localization or quotient.

Let $R$ be a commutative ring, and $p$ be a prime ideal. Let $a_1, ... a_n \in p$. I'm looking for a criterion that the images of $a_1, ... a_n$ in $pR_p/(pR_p)^2$ are spanning or linearly independent,...
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### if R is a commutative ring in which all the maximal ideals are finitely generated then R is Noetherian? [duplicate]

It is well known that if R is a commutative ring in which all the prime ideals are finitely generated then R is Noetherian. Is this also true for maximal ideals instead of primes: if R is a ...
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### Minimal elements under division in a ring

Let $p$ be an element of a commutative ring with unity. The following definition is natural: $p$ is minimal under division if its only divisors (up to equivalence) are $1$ and itself. That is, for ...
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### Show that if a ring $R$ satisfies that for every $x\in R$, $x$ or $1-x$ is invertible then $R$ has a unique max ideal

I tried to prove that if $R$ is like that the group of all not invertible in $R$ is ideal and from there to prove the last but that didnt work for me.
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### If P is a prime ideal of a ring R and P contains no non-zero zero divisors, then R is an integral domain

Here, $R$ is a commutative ring with unity. I realize there are many proofs of this online, but all of them seem to assume a definition of zero divisors that Dummit and Foote does not. Here is what ...
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### Why my proof of every maximal ideal being prime is incorrect

I tried to prove that for a commutative ring $R$ with identity, every maximal ideal $I$ is prime. I think my proof was sort of going in the right direction, but this MSE answer was what I now realize ...
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### Reduced finitely generated k-algebras are isomorphic to $k^n$

We let $k$ denote a fixed algebraically closed field of characteristic zero. We let $R$ denote a reduced finitely generated $k$-algebra where $\dim_kR = n$ as a $k$ vector space and $n$ is a positive ...
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### Prove that if $P$ is a prime ideal of a non-commutative von Neumann regular ring $R$, then $P$ is a maximal ideal

Let $R$ be a ring with identity elements (not necessarily commutative). A ring $R$ is called a von Neumann regular ring if for every $a\in R$ there is $b\in R$ such that $a=aba$. How do I prove that ...
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### Prime Ideals of $\mathbb{Z}\lbrack x \rbrack$
I am looking for a bit of help on part of this problem. Let $R = \mathbb{Z}\lbrack X \rbrack$. For each prime $p \in \mathbb{Z}$, let $I_p$ denote the ideal of $R$ generated by $p$ and $X^2+1$. (a) Is ...
Let $A,B,C$ be commutative rings with given ring homomorphisms $f:A\rightarrow C$ and $g: B \rightarrow C$. Let $A\times_C B:=\{(a,b)\in A \times B : f(a)=g(b)\}$ be their pullback (with the subring ...