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

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2
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1answer
36 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} ...
3
votes
1answer
19 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
25 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.
0
votes
1answer
28 views

How to show if M is projective, then $Tor_i$(M,K)=0, for i>0?

I can't find any clue about proving that, and the so called proof online is too short to understand. Can you offer some clear clues? Thank you!
0
votes
1answer
19 views

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 ...
1
vote
1answer
22 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
24 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 ...
3
votes
1answer
24 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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0answers
16 views

Decomposition of kernel of an epimorphism

Let $f:P→M$ be a projective cover of a left $R$-module $M$ in the sense that $f$ is an $R$-epimorphism from a projective $R$-module $P$ to $M$ with a small kernel in $P$. Assume $P=P_1⊕P_2$ so that ...
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0answers
6 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.
-1
votes
1answer
40 views

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 ...
3
votes
1answer
40 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 ...
2
votes
1answer
54 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 ...
2
votes
1answer
62 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 ...
0
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0answers
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 ...
1
vote
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 ...
4
votes
0answers
65 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 ...
5
votes
1answer
55 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 ...
4
votes
1answer
49 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 ...
2
votes
2answers
53 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 ...
2
votes
2answers
56 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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vote
2answers
63 views

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 ...
1
vote
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 ...
1
vote
1answer
38 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 $M$ ...
0
votes
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 $M$ ...
0
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0answers
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 ...
2
votes
3answers
185 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 ...
0
votes
1answer
32 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 ...
0
votes
1answer
29 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 => ...
3
votes
1answer
77 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 ...
5
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1answer
122 views

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 ...
4
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1answer
115 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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0answers
20 views

Dual object for modules

I am trying to prove the following: $R$ a commutative ring, an $R$-module $M$ has a dual (in the categorical sense) if and only if $M$ is projective and finitely generated. Any hint on how to prove ...
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0answers
31 views

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
52 views

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 ...
0
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0answers
72 views

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
30 views

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 ...
2
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1answer
54 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
57 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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votes
1answer
38 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\\ ...
5
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1answer
69 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 ...
4
votes
2answers
187 views

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 ...
5
votes
1answer
91 views

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
39 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 ...
2
votes
1answer
52 views

Starting projective modules problems

Show that if $n=rs$ where $n,r,s>1$ are positive integers, then the $ \mathbb{Z}_n $-module $r \mathbb{Z}_n$ is projective but it is not free if $(r,s)=1$. Any ideas or help how to prove this ...
0
votes
1answer
46 views

Every finitely generated module is the quotient of a finitely generated projective module.

Every finitely generated module is the quotient of a finitely generated projective module. I already find a proof of this but it uses tensor functor and flat modules propositions so im looking for a ...
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4answers
89 views

Proving that P/PJ is a projective right module over R/J

If P is a projective right module over a ring R and J is a two sided ideal of R. Prove that P/PJ is a projective right module over R/J . My idea was trying to proof that " $M$ is an $R$-module ...
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0answers
54 views

If a direct sum has a projective cover, must the summands have projective covers?

In “Cover of a direct summand” it is asked to show that if a direct sum has a projective cover, and if one of the summands has a projective cover, then so does the other. I gave a solution that works ...
2
votes
1answer
73 views

Cover of a direct summand

Let $L,N$ be $R$-modules. If $L$ and $L \oplus N$ have projective covers, is it true that $N$ admits a projective cover?