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

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1answer
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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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17 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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23 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
53 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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29 views

Projective Resolutions of Finite Type.

Definition. An $R$-module $M$ is said to be of type $FP_n$ if there is a projective resolution $\mathscr{P}$ of $A$ with $P_i$ finitely generated for all $i\leq n$. If the modules $P_i$ are finitely ...
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59 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
26 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
50 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
51 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
22 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\\ ...
4
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1answer
47 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
138 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 ...
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0answers
68 views

When is an $R$-projective module a projective module?

Let $R$ be a semiperfect ring. Is it a true fact that every $R$-projective module $M$ with $Rad(M)$ superfluous in $M$ is projective? I could not reach a good result using just the fact that ...
4
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1answer
81 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
30 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
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1answer
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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 ...
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1answer
39 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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1answer
77 views

Basic short exact sequence exercise in $\mathbb{Z}$-modules [closed]

I was wondering how to properly argue the proof of the following: a) Prove that there is an exact sequence $0 \to \mathbb{Z} \to \mathbb{Z} \to \mathbb{Z}_n \to 0$ (consider $f : \mathbb{Z} \to ...
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4answers
73 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
36 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 ...
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1answer
53 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?
2
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1answer
87 views

Finite projective dimension may lead to projectiveness!

Assume a ring $R$ is injective as an $R$-module. If the projective dimension of an $R$-module $P$ is finite could one conclude that $P$ is a projective $R$-module? Probably one should start with ...
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1answer
58 views

Injective modules in a short exact sequence

Let $0→A→B→C→0$ be an exact sequence in the category of $R$ modules, where $R$ is commutative with $1$, and $B$ be injective. In a text book it is said that all three modules are injective, or the ...
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2answers
59 views

Injective resolution for an integral domain

How could one write an injective resolution for an arbitrary commutative integral domain $R$? Thanks in advance!
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1answer
68 views

if $R$ is a noetherian local ring, then every 2-generated ideal has finite projective dimension iff $R$ is a UFD

This question is about m zcn's comment on my question Projective dimension of all principal ideals is finite. Is R an integral domain?. It's a good point. so i ask it for use of everybody: if ...
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2answers
48 views

The meaning of functor $M \mapsto \mbox{Hom}_A(P,M)$ being exact

(I'm currently studying Lang's text Algebra and it comes up on the page 137. Lang does not explicitly define this expression.) Is the following understanding correct? The function $M \mapsto ...
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1answer
70 views

Finite projective resolution of vector spaces

Let $k$ be a field and $V$ a finite-dimensional vector space. Show that for every natural number $n$ there is a a finite projective resolution of $k$ -vector spaces $0→V_n→V_{n−1} → ··· → V _2 → V _1 ...
3
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1answer
54 views

Analogue of Baer criterion for testing projectiveness of modules

In order to test injectivity of a module $M$ it suffices to check if every linear map from an arbitrary ideal extends to the ring or not. Similarly in order to check the flatness of a module $M$ it ...
4
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2answers
62 views

Why is $(X,Y)$ not a projective $\mathbb{Z}[X,Y]$-module?

This was an exam problem I had which stumped me. The question was to prove that the ideal generated by $X$ and $Y$ in $\mathbb{Z}[X,Y]$ is not a projective $\mathbb{Z}[X,Y]$-module. I was trying to ...
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1answer
38 views

The existence of a projective resolution of M from finite rank free modules

Any R-module admits a projective resolution. When the ring is Noetherian, can we show the existence of a projective resolution of any finitely generated R-module by finite rank free R-modules? In ...
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1answer
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Is there an analogue of Bass-Papp theorem for Projective modules?

The Bass-Papp theorem for injective modules states that If $R$ is a commutative ring such that every direct sum of injective $R$ modules is injective then $R$ is Noetherian. Is there an ...
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1answer
37 views

If a left ideal I of R is an injective R-module then I is a projective R-module. Is the converse true?

I was trying to prove the first statement using some characterizations of injective and projective modules but it did not work out. Could someone drop some hints?
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1answer
37 views

If a left ideal of R is a direct summand of R, is I an injective R-module?

This was an exercise question I encountered in the book on Homological algebra by Vermani. I am unable to prove it or find a counter example.
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0answers
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Why $\mathbb{Q}$ is not a projective $\mathbb{Z}$-module?

From the fact that $\operatorname{Hom}_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})=0$, how do we conclude that $\mathbb{Q}$ is not a projective $\mathbb{Z}$-module?
2
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1answer
91 views

Global dimension of $\mathbb Q [x]$

I'm trying to show that the global dimension of $\mathbb Q [x]$ is $1$. I have shown that $D(\mathbb Q [x]) \leq 1$ as follows. One can reduce to the case of showing that ...
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90 views

Regular sequence and projective module

Let $R$ be commutative ring and $x,y$ an $R$-regular sequence. Then I know that $ R/(x)$ is not a projective $R$-module. My question: Is $R^{2}/(x,y)R$ a projective $R$-module?
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0answers
28 views

Tensoring and retaining projectiveness

Let $A$be a unital associative ring, If $A$ is not a projective $A$-bimodule, however $A\otimes A$ is a projective $A$-bimodule, can we conclude that $A\otimes A \otimes A$ is also projective?
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1answer
38 views

Hom($P$, $R$) $\neq 0 $ if $P$ is a nonzero projective left $R$-module (Rotman)

I've found this exercise, number $3.11$ from Introduction to homological algebra. Prove that $\operatorname{Hom}(P, R) \neq 0 $ if $P$ is a nonzero projective left $R$-module. Any hint?
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2answers
195 views

If $M\otimes N=R^n$ need $M$ be projective?

If over a commutative ring $R$ we have that $M\otimes N=R^n$, $n\neq 0$, need we have that $M$ and $N$ be finitely generated projective? We have finite generation, because if $M\otimes N$ is ...
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1answer
43 views

GCD for multivariable polynomial ring

I'm reading Lectures on Modules and Rings by T. Y. Lam. It's on page 32 of the book, example 2.19A. It reads: (2.19A) Example. Let $k$ be a field. Then in the commutative polynomial ring $R = ...
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0answers
23 views

Differential operators in the language of modules

I am reading articles about differential operators. Authors try to treat differential systems , by studying the differential operator on a vector space in the language of modules. In this manner ...
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1answer
48 views

Exact Sequences of R-Modules

In "A Course in Ring Theory by Passman" it is mentioned, "But the kernel of the combined epimorphism $P\rightarrow B\rightarrow C$ is clearly equal to $E$". I don't understand this part. How can the ...
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1answer
42 views

Annihilator and Projective Dimension

I was reading the book A Course in Ring Theory by Passman and in it is the following lemma; and after this lemma there's a example which I don't quite understand; The main thing that I don't ...
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1answer
23 views

Projective Dimension and Supremum

Here is a lemma that appears in A Course in Ring Theory by Passman. In the last section of the proof the writer shows that, $\mbox{pd }A_i\leq n\iff \mbox{pd }A\leq n$ and finishes the proof. I don't ...
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1answer
51 views

Let $k$ be a division ring, then the ring of upper triangular matrixes over $k$ is hereditary

I'm reading Ring Theory by Louis H. Rowen, and he claimed that The ring of upper triangular matrices over a division ring is hereditary (it's on page 196, Example 2.8.13 of the book). I think it ...
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0answers
64 views

Projective modules over semilocal rings having constant rank are free

I'm starting to study algebraic K-theory by myself and I need a hint how to prove $R$ is a semilocal ring with maximal ideals $\mathfrak m_1,\ldots, \mathfrak m_n$, $P$ is a projective module and ...
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1answer
67 views

Projective and Injective Modules

Let $M$ be a free $\mathbb{Z}$-module. Is $\text{Hom}_{\mathbb{Z}}(M,\mathbb{Q})$ an injective or a projective $\mathbb{Z}$-module? very thanks
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1answer
84 views

The ideal $I=(3,2+\sqrt {-5})$ is a projective module

Let $R=\mathbb Z[\sqrt{-5}]$ and $I=(3,2+\sqrt {-5})$ be the ideal generated by $3$ and $2+\sqrt{-5}$. I'm trying to prove that $I$ is a projective $R$-module. I'm using the lifting property ...
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0answers
71 views

Graded projective modules and vector bundles on projective varieties

Let $S$ be a graded ring which is finitely generated by $S_1$ as an $S_0$-algebra. Let $X = \text{Proj}(S)$. Let $E$ be a vector bundle over $S$. Is $\oplus_{n \in \mathbb{Z}} H^0(X,E(n))$ a graded ...
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Existence of finite projective resolution

The situation I'm considering is quite involved. All rings are noetherian commutative with $1$. All modules are finitely generated. First of all we fix a non reduced local ring $A$ where all zero ...