For questions related to projective modules, their structures, and properties.

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If M and N are nonzero, finitely generated R-modules with M projective, then $M\otimes N$ is nonzero

I am trying to work through the following problem: If M and N are nonzero, finitely generated R-modules with M projective, then $M\otimes N$ is nonzero. My thought on how to approach this problem is ...
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1answer
25 views

Does the nilpotent extension of a $1$-dimensional algebra always give a projective module?

Let $A$ be a $1$-dimensional reduced Noetherian algebra over an algebraic closed field $k$ with characteristic zero. Let $(B,N)$ be a nilpotent extension of $A$, i.e. $B$ is a Noetherian $k$-algebra, ...
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1answer
57 views

example of inverse limit and direct limit

Does a direct limit of projective need to be projective? And is the inverse limit of injectives injective? I guess they need not, but I can't find an example. Can you help please?
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1answer
21 views

generalized inverse in the theory of projective module

A module P over a ring R is projective which is an important topic in the theory of commutative ring due to its structural property of being a direct summand of free module. But my question is why ...
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2answers
43 views

Is $m$ a projective $A$-module?

$A$ is a Noetherian local ring and $m$ be its maximal ideal. Then is $m$ a projective $A$-module? I got this problem while solving another problem. Can anyone please help me to figure it out?
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1answer
96 views

Short exact sequence of modules over a Noetherian local ring of depth $1$.

I am reading an article in algebraic geometry and am having trouble understanding a particular point that reduces to a problem in commutative algebra. I'm not familiar with the concepts involved so am ...
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1answer
42 views

Why is the $\mathbb Z$-module $\mathbb Q$ projective in category $\sigma[\mathbb Q]$?

We know that $\mathbb Q$ is not a projective $\mathbb Z$-module, but, I've read this paper and it says in Example 2.11 that it's easy to check that "$\mathbb Q$ as $\mathbb Z$-module is projective ...
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1answer
20 views

$P$ is projective . Show that $P$ is a direct summand of a free $R-$ module

$P$ is projective . Show that $P$ is a direct summand of a free $R-$ module I am using the definition of a projective module as $P$ is projective if every exact sequence $M\rightarrow P\rightarrow ...
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1answer
39 views

“Direct sums of injective modules over Noetherian ring is injective” and its analogue

I have a commutative algebra class and I heard the theorem from the professor: Let $R$ be a Noetherian ring and $\{E_i : i\in I\}$ be a collection of injective $R$-modules then $\bigoplus_{i\in I} ...
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1answer
29 views

Two short exact sequences with projective objects in the middle

Problem: Prove that for two short exact sequences $$ 0\rightarrow A \xrightarrow{f} B \xrightarrow{g} C \rightarrow 0 $$ $$ 0\rightarrow A' \xrightarrow{f'} B' \xrightarrow{g'} C \rightarrow 0, $$ ...
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1answer
28 views

Projective resolution of ideal $\langle a,b\rangle $ in the $K=R[a,b]$

I saw a resolution as $$K\to K^2 \to \langle a,b\rangle \to 0,$$ but I can't figure out why, and can't figure out the maps. Could you give me some ideas? Thank you.
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1answer
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A question about the quotient of two chain complexes of $R$-modules

Let $P$ be an acyclic object of $Ch_R$, let $P^{(k)}$ be the chain subcomplex of $P$ which agrees with $P$ above the degree $k-1$, contains $Bd_{k-1}P$ in degree $k-1$, and vanishes below degree ...
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1answer
24 views

$\text{Coker}(f),\text{Coker}(g)$ projective modules $\Rightarrow$ $\text{Coker}(gf)$ projective

Let $\textbf{Ch}_R$ the category of chain complexes of $R-$modules ($R$ is an associative ring with unit). I want to prove that this cat. satisfies the model category axioms. In particular we want to ...
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0answers
25 views

Connection between Chladni Plates and Projective Geometry?

Does anybody know of a connection between Chladni Plates and Projective Geometry? Any published research on the subject? I am interested to know if Chladni Plates (or surfaces) can be ...
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29 views

Equivalent properties of projective modules

For a left $R$ module $P$, the following are equivalent: Given $M\xrightarrow{\psi} N \to 0$ exact and $\omega: P \to N$, there exists $\tilde{\omega}: P \to M$ such that $\psi \circ ...
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7 views

can we decompose every module as a direct sum of projective submodule P and a submodule C such that C has no nonzero projective submodule

In the noetherian rings, we can write every module as a direct sum of injective submodule E and a submodule D where D has no injective submodule.
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Projective dimension of module over local ring

This question arose reading the well known article by Buchsbaum Lectures on regular local rings. He states without proof that, given $(R,m)$ a local ring and an $R$-module $M$ over $R$, we have the ...
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1answer
43 views

Semisimplicity of a ring

As it is well-known, a ring with unity $R$ is semisimple if and only if each left $R$-module is projective. My question: Is simisimplicity of $R$ equivalent to each "simple" left $R$-module being ...
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1answer
62 views

Characterization of projective and injective modules

In Theorems 4.7 and 8.4 Hilton & Stammbach give two lists of 5 different characterizations of projective and injective modules, respectively. Even though I can follow the proofs they give, I'd ...
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1answer
71 views

Hartshorne II Ex 5.9(a) or R. Vakil Ex 15.4.D(b): Saturated modules

The question is basically like this: Prove that if $S_{\cdot}$ is a finitely generated (in degree 1) graded ring over a field $k$ and $M_{\cdot}$ is finitely generated, then the saturation ...
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32 views

Projectivity of a certain module

Let $M$ be a generator for the category of left $R$-modules, and let we have an $R$-epimorphism $h$ from $R^{(X)}$ to an $R$-module $P$ which is projective relative to $M^{(X)}$ ($X$ is a set). I want ...
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1answer
30 views

Projective modules over enveloping algebra

How could I prove the following statement? Let $k$ a commutative ring and let $A$ an associative $k$-algebra that is also a projective $k$-module. Then every projective $A\otimes A^{op}$ left module ...
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0answers
66 views

Bass' paper on Gorenstein rings

I am currently reading the paper On the ubiquity of Gorenstein rings by Hyman Bass. I found difficulty to understand the proof of Proposition (7.2). Under the the following setting: $A$: commutative ...
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1answer
62 views

A free direct sum of a projective module

I want to prove that a left module $_RP$ is a projective generator if and only if a direct sum of (copies of) $P$ is free. My try is, first, to observe that $P$ is a generator if and only if for ...
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1answer
55 views

Picard group of $\mathbb Z[\sqrt{-5}]$

I search for a simple proof for the fact that $\operatorname{Pic}(\mathbb Z[\sqrt{-5}])=\mathbb Z/2\mathbb Z$, where $\operatorname{Pic}(R)$ is the Picard group of the ring $R$ - the set of ...
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2answers
55 views

A non-projective module

Let a ring with identity $R$ be decomposed as $S_1⊕\cdots⊕S_n$ (as a right $R$-module), where $S_i=e_iR$ with $e_i$ nonzero idempotents of $R$ adding up to $1$. If $J$ is the Jacobson radical of ...
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2answers
60 views

Are projective modules “graded projective”?

Let $A^{\bullet}$ be a graded commutative algebra. Denote by $A^{\bullet}$-mod the category of graded modules over $A^{\bullet}$. Let $A$ be $A^{\bullet}$ considered as an algebra (we forgot grading). ...
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2answers
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Trouble on Aluffi's Exercise 5.5, Chapter 6: finitely generated projective module over a local ring is free

The overall goal of the problem is the following: Let $R$ be a local commutative ring (so that there is a unique maximal ideal $\mathfrak{m}$) and let $M$ be a finitely generated $R$-module. Assume ...
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1answer
37 views

$pd(M) \leq n$ implies $\ker(P_n \to P_{n-1})$ projective

Let $M$ be a finitely generated $A$-module with $A$ Noetherian. Suppose $pd(M) \leq n$. Then given any projective resolution $$\ldots \to P_n \to P_{n-1} \to \ldots \to P_0 \to M \to 0$$ why is the ...
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1answer
42 views

Is a projective $R[G]$-module a projective $R[H]$-module if $H$ is a subgroup of $G$?

I have a ring $R$ of characteristic $0$ and a finite group $G$. Let $H$ be a subgroup of $G$. Question: If $M$ is a projective $R[G]$-module where $R[G]$ is the usual group ring then is $M$ ...
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1answer
31 views

Inversion of an element in Picard group over commutative ring

I'm having some troubles understanding a proof in Commutative Algebra Chapter I - VII of N. Bourbaki. It's on pag 114 of the book. Here's what it says: Theorem 3 ... (ii) Conversely, if $M$ ...
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53 views

Non finitely-generated projective $\mathbb{Z}$-module [duplicate]

Let $M$ be a projective $\mathbb{Z}$-module. Must $M$ be free? It is easy to see that the answer is yes if $M$ is finitely generated, but I do not know about the general case. If the answer ...
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3answers
209 views

Is every submodule of a projective module projective?

Is every submodule of a projective module projective? I know that the answer is no, but I haven't been able to come up with any concrete examples despite quite a bit of effort. Also, if the ...
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1answer
33 views

simple projectile motion problem solving equation

a stone is thrown with a velocity of 20m/s at an elevation of angle A, given by tan A = 3/4, what horizontal distance does it cover in 2 sec, and what is its height then above the horizontal plane ...
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1answer
30 views

projectile motion (dynamic) stone

2) A stone is projected downwards with a velocity of 20m/s at an angle of 30degrees below a horizontal line through the point of projection. Find the velocity of the stone after 2 sec. t=0 => ...
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1answer
97 views

Is a stably free module always free? [duplicate]

Let $R$ be a commutative ring with unity and $M$ an $R$-module. If $M\oplus R^m\cong R^n$ for some integers $m,n \geq 1$ then must $M$ be finitely generated and free? Can somebody help me with ...
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Generic freeness: $M$ f.g. projective, then there is $a\notin \mathfrak p$ for which $M[a^{-1}]$ is a free $R[a^{-1}]$-module.

In Jacobson's BAII, he aims to show that any finitely generated projective module over a connected ring has a rank, where he defines this as follows: First, he shows that any finitely generated ...
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116 views

Rank of projective module defined as the smallest $n$ such that $P$ is a direct summand of $R^n$

Over a commutative ring $R$, the rank of a projective module $P$ is defined by looking at the map $\text{rank}(P) : \text{Spec}(R) \rightarrow \mathbb{N}_0$ given by $\mathfrak{p}\mapsto ...
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Confused with a terminology used in a book Group Representation Vol 1 Part A by G Karpilovsky

I am confused with a terminology used in a book Group Representation Vol 1 Part A by G Karpilovsky (Chapter 10, page 441). Which is as follows: Let $A=\oplus_{g \in G} A_g$ be a $G$-graded ...
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1answer
62 views

A question about projective modules.

Suppose that we have a commutative ring $R$ with an idempotent $e$, and $M$ an $R$-module such that $Me$ is $Re$-projective. I am interested to know under which conditions this implies that $M$ is ...
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0answers
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Gorenstein ring and projective module

I am new to this topic and would appreciate little explanation. Def: A commutative, unital ring $A$ is a cubic ring if $A$ is a free $\mathbb{Z}$-module of rank $3$. Def : A cubic ring $A$ is ...
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Algebraic K-theory: Categories of modules and their equivalences

I'm currently preparing a presentation on the second chapter, called "Categories of modules and their equivalences", in Algebraic K-theory by Hyman Bass. I have a VERY elementary understanding of ...
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1answer
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Projective Spaces which are not Vector Spaces

I'm studying Projective Spaces, I've collected a few books and most of them define Projective Spaces in terms of Vector Spaces, that is, they define a 'projective space structure" in the vector space ...
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56 views

Is the coefficient ring $R$ of a group ring $RG$ necessarily projective as an $RG$-module?

So this may be a trivial question but I am new to the idea of group rings. Suppose we have a ring $R$ and a group $G$, I was wondering if the trivial $RG$-module $R$ is projective? In which case, how ...
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1answer
58 views

A condition of equivalence of flatness and projectiveness

This is a problem in "Foundations of Module and Ring Theory" of Wisbauer: " Let $R$ be a subring of the ring $S$ containing the unit of $S$. Show that a flat $R$-module $N$ is projective if and only ...
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1answer
39 views

an ideal of matrix ring which is projective

Let $K$ be a field and $$ A=\left\{ \begin{pmatrix} a&b&c\\ d&e&f\\ 0&0&g \end{pmatrix} :a,\dots,g\in K \right\}, $$ then $$ J=\left\{ \begin{pmatrix} 0&0&c\\ ...
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1answer
76 views

Projective modules over $kG$ equivalent to injective.

Let $k$ be a field and $G$ is finite group. I want to prove that a $kG$ module $P$ is projective iff it's injective. I proved that if module is projective then it's injective. 1) $kG$ is injective ...
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2answers
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Proving that tensoring a projective module with a flat module gives a projective module?

If $P$ is a projective module and $M$ is a flat module, both over some commutative ring $R$, then how do you prove that $M\otimes_R P$ is flat? I tried using the fact that $P$ is the direct ...
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1answer
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When is the rank the biggest number for which $\Lambda^m(M) \neq 0$?

I was doing some theory of Dedekind domains, and I found very useful to use the language of exterior algebra to prove the main results for finitely generated modules over Dedekind domains. I was, ...
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1answer
40 views

Morita equivalence: Is $_{\mathrm{End}_R(P)}P$ projective if $P_R$ is?

Assume $P$ is a right projective $R$-module. Is $P$, viewed as a left $\mathrm{End}_R(P)$-module, projective as well? If not, under what conditions does it hold? Context: I am trying to ...