# Tagged Questions

Questions about commutative rings, their ideals, and their modules.

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### Modules as morphisms to endomorphism rings

An $A$-module $M$ may be thought of as a (surjective) ring homomorphism $f: A \to E(M)$, where $E(M)$ is a ring of group endomorphisms of $M$. Then $am = f(a)(m)$. Is there any more to this ...
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### Irreducible elements for a commutative ring that is not ID

Why does the definition of an irreducible element require us to be in an integral domain? Why can we not define an irreducible element exactly the same in a commutative ring that is not an integral ...
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### Finitely generated projective modules over polynomial rings with integral coefficients

There is famous Quillen-Suslin theorem which states that every finitely generated projective module over a ring of polynomials $k[x_1,...,x_n]$, where $k$ is a field, is free. I have never carefully ...
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### How do I find the ideal $I+J$?

This is a homework problem: Consider the polynomial ring $R=\mathbb Z_2[x_0,x_1,\dots,x_n]$. Let $I=\langle x_0x_1\cdots x_n\rangle$ and $J=\langle x_0+x_1,x_0+x_2,\dots,x_0+x_n\rangle$. Find $I+J$...
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### What properties are preserved by direct limits? [on hold]

We know that direct limit of a directed family of flat $R$-modules is also flat ($R$ is a commutative ring with $1$ and all modules are unital). I am looking for other properties of modules which ...
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### In an $\Bbb{N}$-graded domain $A$, units are homogeneous

Let $A$ be a graded domain, with additive subgroups $A_n,\,\forall\,n\geq 0$, s.t. ${A_n\cdot A_m}\subseteq A_{n+m}\,\forall\,n,m\geq 0$, and $A=\bigoplus_{n=0}^\infty\, A_n$ as abelian groups. I wish ...
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### A prime ideal which is not maximal

I am searching for a prime ideal of the ring $R=∏_{n=2}^{∞} {\mathbb Z}_{2^n}$ which is not maximal. In fact, since each ${\mathbb Z}_{2^n}$ is local with $\left<\bar 2\right>$ as the maximal ...
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### How do you find the free resolution of the module $M$ and of $F/M$ where $F=(K[x,y])^3$?

$M$ is a module generated by $$f_1=(xy,y,x), f_2=(x^2+x,y+x^2,y), f_3=(-y,x,y),f_4=(x^2,x,y).$$ We're to use the lex ordering with $x<y$ and $e_1>e_2>e_3$, where terms are given preference ...
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### Help finding an article [on hold]

Hello Recently I have been studying algebra and am in search of the following paper : Kac, V. G. Classification of simple $Z$-graded Lie superalgebras and simple Jordan superalgebras. Comm. Algebra 5 ...
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### How to decompose that ideal?

We have $$I=\left(x^2+2y^2-3,y(x-y),y(y+1)(y-1)\right)\subset\mathbb{C}[x,y]$$ and I would like to decompose it as intersection of simpler ideals. How could I proceed? For example, in this ...
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### Extension of Scalars is well-defined

The reason I'm asking this, is because as an exercise, I'm asked to prove the following: Let $A$, $B$ be rings, $f:A\to B$ a ring homomorphism inducing $A$-module structure on $B$, and $M$ a flat $A$-...
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### Checking regularity via completion

It is well-known that a local Noetherian ring $A$ is regular if and only if its completion is regular, and that one can check (if, say, $A$ is a $k$-algebra) by observing that it is a power series ...
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### If some polynomial is in an ideal $I$, how can I write it as a linear combination of the generators of $I$?

I'm looking for a (easy) procedure of some sort. I also know a little bit of Singular and CoCoA, and was wondering if you can do that in there?
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### Definition of singular points on an algebraic curve

From what I understood, given a point $p$ on a scheme $X$ over a field $k$, we have \dim \mathcal{O}_{X,p} \leq \dim_{\mathcal{O}_{X,p}/\mathfrak{m} }\mathfrak{m}/\mathfrak{m}^2 \end{...
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### Structure constants in a finitely generated $\mathbb{k}$-algebra

Let $\mathbb{k}$ be a field of characteristic $0$. Suppose we have a finitely generated graded $\mathbb{k}$-algebra $A= \bigoplus_{i=0}^{\infty}A_i$ which is free of finite rank as a module over a ...
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### Finding an essential submodule

Let $R$ be a commutative ring with unity, and let $M$ be a unitary faithful $R$-module. Assume the annihilator $A$ of $r\in R$ is essential in $R$ (as an $R$-module). I search for an essential ...
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### Specific basis of A-algebra B that is also a free A-module of finite rank.

I have a problem that seems (at least to me) harder then I initially thought. Let $B$ be an $A$-algebra that is also a free $A$-module of finite rank (if necessary we can assume that $B$ is ...
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### Is an algebra homomorphism between two finitely generated algebras over a field automatically an integral morphism?

I'm having a bit of trouble with the idea of an integral morphisms, and algebra homomorphisms for that matter. I'm wondering if the above is just "automatically" true. Does an algebra over a field ...
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### relation between the Krull dimension and the dimension of vector spaces

Let $(R, \frak m)$ be a commutative Noetherian local ring, and $M$ be an $R$-module such that $\frak m$$M =0$. We know that $M$ can be considered as a $R/ \frak m$-module, namely as a vector space on ...
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### If $R\otimes_\mathbb R\mathbb C$ is finitely generated $\mathbb C$ - algebra then $R$ is a finitely generated $\mathbb R$ - algebra?

Let $R$ be an $\mathbb R$ - algebra. Suppose $A=R\otimes_\mathbb R\mathbb C$ is a finitely generated $\mathbb C$ - algebra then is $R$ a finitely generated $\mathbb R$ - algebra? I thought along the ...
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### polynomial ring with isomorphic quotients

If $R$ is a commutative ring and $f(x), g(x) \in R[x]$ two polynomials such that $R[x]/f(x)\cong R[x]/g(x)$ as $R$-algebras, what can we say about $f$ and $g$? Or given $f(x)\in R[x]$, what can we ...
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### Dimension of polynomial rings and tensor products of residue fields

In Matsumura textbook to show that $\dim A[x] = \dim A + 1$, first it states that $A[x] \otimes k(\mathfrak{p}) = k(\mathfrak{p})[x]$ which is one dimensional. Then it uses the theorem 15.1.(ii) since ...
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### primary decomposition of injective envelope of a module

The Exercise A3.6 of Eisenbud's book, Commutative Algebra with a view Toward Algebraic Geometry, is: Assuming that $R$ is Noetherian, let $M$ be any finitely generated $R$-module. a. Let ...
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### How can I proceed to find a maximal principal ideal in $\mathbb Z_{(2)}[x]$?

How can I proceed to find a maximal principal ideal in $\mathbb Z_{(2)}[x]$? I know the answer in the sense that i know that $(2x+1)$ is a maximal principal ideal of that polynomial ring. But if i ...
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### Is there a commutative ring with a “generalized determinant”?

Does there exist a commutative ring(-with-a-1) $R$ and positive integer $n$ and function $\hspace{.04 in}f$ from [the set of $n$-by-$n$ matrices over $R$] to $R$ such that $f$ is linear in each row ...
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### Tensor products and Residue fields

Given a ring homomorphism between two Noetherian rings, $f:A \to B$. Let $P$ be a prime ideal in $B$ and let $\mathfrak{p}$ be an ideal in $A$ such that $f^{-1}(P) = \mathfrak{p}$. How can we prove ...
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### Localization at associated prime of a principal ideal and ideal generator [on hold]

Let $A$ be a commutative Noetherian local ring and $I=(a)$ a principal ideal of $A$. Let $P$ be an associated prime of $A/I$. Is $a$ a maximal regular sequence on $A_P$ (i.e., $a$ is not a zero ...
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I want to prove that the polynomial ring $k\left[x,y\right]$ is not integral over the subring $k\left[xy,y\right]$ , where $k$ is a field. My claim is that $x$ is not integral over $k\left[xy,y\... 1answer 57 views ### Rings of Krull dimension one I have to write a monograph about commutative rings with Krull dimension$1$, but I can't find results, so I am looking foward for some references, and some results to search. Also, I would appreciate ... 1answer 33 views ### Induced homomorphism on Spectra of rings In Matsumura textbook, there is this following statement. A ring homomorphism$f:A \to B$, induces a map$f': \operatorname{Spec}B \to\operatorname{Spec}A$under which an element$\mathfrak{p} \...
Consider the following situation. Let $k$ be a characteristic $0$ field, and consider an étale morphism of $k$ schemes $f:X\rightarrow Y$. Moreover, let $K$ and $L$ be two extension fields of $k$ such ...