$rad(P)$ non-projective induces all submodules of P non-projective?
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
Let $A$ be a finite dimensional algebra. $P$ is an idecomposable projective $A$-module such that its radical $rad(P)$ is non-projective.
Is it right that every non-zero proper submodule of $P$ is not projective?
modules representation-theory
add a comment |Â
up vote
0
down vote
favorite
Let $A$ be a finite dimensional algebra. $P$ is an idecomposable projective $A$-module such that its radical $rad(P)$ is non-projective.
Is it right that every non-zero proper submodule of $P$ is not projective?
modules representation-theory
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $A$ be a finite dimensional algebra. $P$ is an idecomposable projective $A$-module such that its radical $rad(P)$ is non-projective.
Is it right that every non-zero proper submodule of $P$ is not projective?
modules representation-theory
Let $A$ be a finite dimensional algebra. $P$ is an idecomposable projective $A$-module such that its radical $rad(P)$ is non-projective.
Is it right that every non-zero proper submodule of $P$ is not projective?
modules representation-theory
asked Aug 11 at 3:05
Xiaosong Peng
639414
639414
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
No. Consider $A$ the quotient of the path algebra of the square (i. e. 4 vertices and arrows $a: 1to 2, b: 2to 4, c: 1to 3, d:3to 4$) modulo the ideal spanned by $ba-dc$. Then $operatornamerad (P_1)$ is not projective while $P_2, P_3, P_4$ are all submodules of $P_1$.
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
No. Consider $A$ the quotient of the path algebra of the square (i. e. 4 vertices and arrows $a: 1to 2, b: 2to 4, c: 1to 3, d:3to 4$) modulo the ideal spanned by $ba-dc$. Then $operatornamerad (P_1)$ is not projective while $P_2, P_3, P_4$ are all submodules of $P_1$.
add a comment |Â
up vote
2
down vote
accepted
No. Consider $A$ the quotient of the path algebra of the square (i. e. 4 vertices and arrows $a: 1to 2, b: 2to 4, c: 1to 3, d:3to 4$) modulo the ideal spanned by $ba-dc$. Then $operatornamerad (P_1)$ is not projective while $P_2, P_3, P_4$ are all submodules of $P_1$.
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
No. Consider $A$ the quotient of the path algebra of the square (i. e. 4 vertices and arrows $a: 1to 2, b: 2to 4, c: 1to 3, d:3to 4$) modulo the ideal spanned by $ba-dc$. Then $operatornamerad (P_1)$ is not projective while $P_2, P_3, P_4$ are all submodules of $P_1$.
No. Consider $A$ the quotient of the path algebra of the square (i. e. 4 vertices and arrows $a: 1to 2, b: 2to 4, c: 1to 3, d:3to 4$) modulo the ideal spanned by $ba-dc$. Then $operatornamerad (P_1)$ is not projective while $P_2, P_3, P_4$ are all submodules of $P_1$.
answered Aug 11 at 4:53
Julian Kuelshammer
7,12632564
7,12632564
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2879006%2fradp-non-projective-induces-all-submodules-of-p-non-projective%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password