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Questions tagged [projective-module]

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

We call a module $P$ over a ring $R$ projective if for every surjective $R$-module homomorphism $f : N \to M$, and every module homomorphism $g : P \to M$, there is a lift $h$. That is, there is a module homomorphism $h : P \to N$ with $fh = g$.

Alternatively, a module $P$ is projective if every short exact sequence $$0 \to A \to B \to P \to 0$$

of $R$-modules splits.

Projective modules can be viewed as generalizations of free modules, and every free module is projective.

Source: Projective module.

790 questions
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Let A be a submodule of a projective R-module B, show that A is projective if B/A is projective

I want help to the following problem(especially I want to tell me which is the basic idea behind the solution): Problem: We have the following assumptions: Let $B$ is projective $R-$module $A$ is $R-$submodule of $B$ and $B/A$ is projective…
TrItOs
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For a finite group $G$ and field $k$ of char$=p$, if $P,P'$ are projective $k[G]$-modules with $[P]=[P']$, is it true that $P=P'$?

That is -- is it true that if projective $k[G]$-modules have same composition factors then they are isomorphic? This is easy to see for $\text{char}(k)=0$, or if $G$ is a composition of a $p$-group and a $p'$-group. Serre in "Linear Representations…
George
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(Zelmanowitz) Regular Module but not Projective

An $R-module$ $M$ is called regular (Zelmanowitz) if given any $m \in M$, there exists R-module homomorphism $f:M \longrightarrow R$ such that $m=f(m)m$. I have a problem to find a regular module but not projective. I had proof that any submodule of…
Valentino
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direct sum of projective module is free

Would anyone explain how the direct sum of projective modules are free (over a commutative ring)? What I know is an R-module P is said to be projective if to every surjective homomorphism $\alpha : B \rightarrow C $ of R-modules and to every…
Amanda
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If $a^{3}=a$ for some non-zero element $a$ in $R$, then the ideal $Ra$ is projective $R$-module

I need help to the following problem: Problem: Let $R$ be a commutative ring with $1 \neq 0$. Let $a$ be a non-zero element of $R$ such that $a^3=a$, then the ideal $Ra$ is a projective $R$-module. Solution(my attempt): We have that $a^{3} = a…
TrItOs
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Projective resolution of $\mathbb{Z}/p^n\mathbb{Z}$ on $\mathbb{Z}/p^k\mathbb{Z}$

Consider $R=\mathbb{Z}/p\mathbb{Z}$ for $p$ prime and $n>1$ as ring and $M_k=\mathbb{Z}/p^k\mathbb{Z}$ as $R$- module ($k \leq n$). I want to find a projective resolution of $M_k$. $R$ is projective as $R$- module and my idea is to use a projective…
Mario
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$UP = _fP$ is a projective $\mathbb{Z}S$-module.

$S$ is a subgroup of $G$ and f is the inclusion map. As an $\mathbb{Z}S$-module, $\mathbb{Z}G$ is free. It follows that if $P$ is a projective $\mathbb{Z}G$-module, then $UP = _fP$ is a projective $\mathbb{Z}S$-module. Here $U$ is the change of…
scsnm
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R is projective Q-module?

The set of real numbers $\mathbb{R}$ is project $\mathbb{Q}$-module? I think it is not but I cannot prove it. How can I prove it?
junun
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for an idempotent linear map T, Range T is projective

Let A be an $m \times n$ regular matrix of rank k. Consider $A$ as a module homomorphism from $\mathbb{R}^n$ into $\mathbb{R}^m$. Since $A$ is regular, there exists a matrix $G:\mathbb{R}^m \rightarrow \mathbb{R}^n$ such that $AGA = A$. Now $AG$ is…
Amanda
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Show that $\mathbb{Z}_2$ is not a free $\mathbb{Z}_6$-module

Let $ \Lambda = \mathbb{Z}_6 $，the ring of integers modulo $ 6 $.Since $ \mathbb{Z}_6 = \mathbb{Z}_2 \oplus \mathbb{Z}_3 $ as a $\mathbb{Z}_6 $-module,then $\mathbb{Z}_2$ as well as $\mathbb{Z}_3$ are projective $\mathbb{Z}_6$-modules.How ,they are …
unicornki
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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 through the point of projection? solution: r(right)…
sekling
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