# Tagged Questions

56 views

### Example of Localization and Prime Ideals

For each $n\in \mathbb Z^+$, give an example of a localization of $\mathbb Z$ with exactly $n$ prime ideals. Justify your answer. Could an example have something to do with a UFD or Noetherian ...
48 views

### Example of $\sum_i a_i\otimes b_i\in M\otimes_AN$ which cannot be written as $a\otimes b$

In the appendix of my commutative algebra text: Note that in general the element of $M\otimes_AN$ is a sum of the form $\sum_i a_i\otimes b_i$ and cannot be necessarily written as $a\otimes b$. ...
15 views

### If $b+(a)$ is not a zero divisor in $R/(a)$, does it follow that $(a,b)=R$

Let $R$ be a commutative ring with identity. Let $a,b$ be elements of $R$. If $b+(a)$ is not a zero divisor in $R/(a)$, does it follow that $(a,b)=R$ ? The converse can be easily shown to be true. ...
223 views

### Example of non-noetherian algebras which are tensor products of noetherian algebras

We suppose all rings are commutative with unity. I am looking for examples of a tensor product $B\otimes_A C$ which is not noetherian, where $A$ is a noetherian ring and $B, C$ are noetherian ...
180 views

### Counterexamples to Nakayama's Lemma if $M$ is not finitely generated

One of the most famous forms of Nakayama's lemma says: Let $I$ be an ideal in $R$ and $M$ a finitely-generated $R$ module. If $IM = M$, then there exists an $r \in R$ with $r ≡ 1 \pmod I$, ...
222 views

### Examples of non-isomorphic fields with isomorphic group of units and additive group structure

YACP mentions in a comment that: There are examples of non-isomorphic fields $K$ and $L$ with $(K,+)\cong (L,+)$ and $(K^{\times} ,\cdot)\cong (L^{\times},\cdot)$ Can someone provide an ...
165 views

### Existence of minimal non-zero prime ideals: Counter examples?

Let $R$ be an integral commutative ring with unit. If $R$ is noetherian, then every ideal has finite height, in particular, there exist minimal non-zero prime ideals if (and only if) $R$ is not a ...
185 views

### Example of rings of the same positive characteristic that do not embed into their tensor product?

I'm overcoming my fear of tensor products, and the following exercise got me wondering: Give an example of commutative rings $A$ and $B$ with $\operatorname{char}A=\operatorname{char}B$ such that ...
91 views

### Can we find a subset of $Spec(R)$ not quasi-compact?

If $R$ is a commutative ring with unit, we can easy prove that $Spec(R)$ is quasi-compact. However can you give me an example of $R$ such that a subset $A \subset Spec(R)$ isn't quasi-compact?
65 views

### How can I calculate $\mathrm{Spec}(\mathbb{Z}_{(3)})$? And $\mathrm{Spec}(\mathbb{Z}_3)$?

Let $\mathbb{Z}_{(3)}$ be the localization (in $\mathbb{Z}$) of the ideal generated by $3$. So I have to put in $\mathbb{Z}$ all the inverses of the complement of $(3)$. How can I calculate ...
539 views

### Free modules over commutative rings. [duplicate]

Free modules over a commutative ring $R$ with $1$ have well-defined rank. I have been wondering if there is a ring $R$ such that there are free modules $M'\subset M$ with ...
269 views

### Extending Herstein's Challenging Exercise to Modules

Anybody who has worked through Herstein's Topics in Algebra might remember Exercise 26 of Section 2.5 (in second edition): If $G$ is an abelian group containing subgroups of order $m$ and $n$, ...
82 views

### Integral extensions of rings, when one of the rings is a field

The following is from page 61 of Introduction to Commutative Algebra by Atiyah & Macdonald: Proposition 5.7. Let $A\subseteq B$ be integral domains, $B$ is integral over $A$. Then $B$ is a ...
115 views

### A domain with only a (non-zero) prime ideal

What is an example of a domain $A$ such that Spec$A=\{(0),\mathfrak p\}$? For instance one could find a principal ideal domain that is also a local ring but I can't imagine such a ring.
267 views

### Injectivity of Homomorphism in Localization

Let $\alpha:A\to B$ be a ring homomorphism, $Q\subset B$ a prime ideal, $P=\alpha^{-1}Q\subset A$ a prime ideal. Consider the natural map $\alpha_Q:A_P\to B_Q$ defined by ...
213 views

### Is the integral closure of a Henselian DVR $A$ in a finite extension of its field of fractions finite over $A$?

This question is related to the one here: A question related to krull akizuki In the answers to that question, some examples are given of a discrete valuation ring $A$ and a finite (necessarily ...
263 views

### Deducing results in linear algebra from results in commutative algebra

Here are two examples of results which can be deduced from commutative algebra: Any $n\times n$ complex matrix is conjugate to a Jordan canonical matrix (can be proven using the structure theorem ...
226 views

### If $R$ is commutative, and $J\lhd I\lhd R,$ does it follow that $J\lhd R?$

$\lhd$ will stand for "is an ideal of" in this post. Let $R$ be a commutative ring, $J\lhd I\lhd R$. Does it follow that $J\lhd R?$ I don't think it does, but I'm having difficulty finding a ...
806 views

### A free submodule of a free module having greater rank the submodule

Let $R$ be a commutative ring, and let $N\leq M$ be $R$-modules. Then, suppose $M$ and $N$ are free over $R$, if $R$ is an integral domain, then -considering the fraction modules over the quotient ...
197 views

### Example of height $n$ ideal with $I/I^2$ (locally) $n$-generated, but $I$ is not.

For $R$, a commutative noetherian ring of dimension $d$, I'm looking for an example where $I \subset R$ is an ideal of height $n \lt d$ such that $I/I^2$ is generated by $n$ elements (locally ...
308 views

### Non-Noetherian rings with an ideal not containing a product of prime ideals

It is well-known that in every commutative Noetherian ring every ideal contains a product of prime ideals. Are there examples of non-Noetherian rings with an ideal that does not contain any prime ...
88 views

### Azumaya algebra and its subalgebras

I remind you that an Azumaya algebra $A$ is a central and separable algebra. Now, I know that if $A$ is an algebra over a skew-field or over a local ring then there exists a subalgebra $S$ of $A$ such ...
1k views

### examples of faithfully flat modules

I'm studying some results about flatness and faithful flatness and I'd like to keep in my mind some examples about faithfully flat modules. In general, free modules are the typical example. Another ...
I am looking for an example of an exact sequence of $R$-modules $$0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$$ and a $R$-module $N$, such that  0 \rightarrow M' ...
I'm trying to think of an example of a homomorphism of commutative rings $f:A\rightarrow B$ and ideals $I,J$ of $B$ such that $f^{-1}(I)+f^{-1}(J)$ is not a preimage of any ideal of $B$. I can't seem ...