Module epimorphism and homomorphism between quotients of radical series

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I am reading Alperin's Local Representation Theory. On page 37, there goes an inference like this:



$P, Q$ are $A$-modules. $Q$ is a homomorphic image of $P$. Then $mathrmrad Q/ mathrmrad^2 Q$ is a homomorphic image of $mathrmrad P/ mathrmrad^2 P$.



The radical refers to Jacobson radical. $mathrmrad^2 M=mathrmrad(mathrmrad M)$.



All the modules or algebras here are finite dimensional over some field $k$. A little more properties is assumed but seems not considered. Such as $Q$ is uniserial with length $2$, with composition factors isomorphic to $k$.



I wonder why this is true. An epimorphism may not send radical onto radical. By definition, the quotients are semisimple. It seems to say that all the simple submodules to appear in $mathrmrad Q/ mathrmrad^2$ must appear in $mathrmrad P/ mathrmrad^2 P$. But I have no idea how to show this.






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    I am reading Alperin's Local Representation Theory. On page 37, there goes an inference like this:



    $P, Q$ are $A$-modules. $Q$ is a homomorphic image of $P$. Then $mathrmrad Q/ mathrmrad^2 Q$ is a homomorphic image of $mathrmrad P/ mathrmrad^2 P$.



    The radical refers to Jacobson radical. $mathrmrad^2 M=mathrmrad(mathrmrad M)$.



    All the modules or algebras here are finite dimensional over some field $k$. A little more properties is assumed but seems not considered. Such as $Q$ is uniserial with length $2$, with composition factors isomorphic to $k$.



    I wonder why this is true. An epimorphism may not send radical onto radical. By definition, the quotients are semisimple. It seems to say that all the simple submodules to appear in $mathrmrad Q/ mathrmrad^2$ must appear in $mathrmrad P/ mathrmrad^2 P$. But I have no idea how to show this.






    modules







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      up vote
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      up vote
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      down vote

      favorite











      I am reading Alperin's Local Representation Theory. On page 37, there goes an inference like this:



      $P, Q$ are $A$-modules. $Q$ is a homomorphic image of $P$. Then $mathrmrad Q/ mathrmrad^2 Q$ is a homomorphic image of $mathrmrad P/ mathrmrad^2 P$.



      The radical refers to Jacobson radical. $mathrmrad^2 M=mathrmrad(mathrmrad M)$.



      All the modules or algebras here are finite dimensional over some field $k$. A little more properties is assumed but seems not considered. Such as $Q$ is uniserial with length $2$, with composition factors isomorphic to $k$.



      I wonder why this is true. An epimorphism may not send radical onto radical. By definition, the quotients are semisimple. It seems to say that all the simple submodules to appear in $mathrmrad Q/ mathrmrad^2$ must appear in $mathrmrad P/ mathrmrad^2 P$. But I have no idea how to show this.






      modules







      share|cite|improve this question















      I am reading Alperin's Local Representation Theory. On page 37, there goes an inference like this:



      $P, Q$ are $A$-modules. $Q$ is a homomorphic image of $P$. Then $mathrmrad Q/ mathrmrad^2 Q$ is a homomorphic image of $mathrmrad P/ mathrmrad^2 P$.



      The radical refers to Jacobson radical. $mathrmrad^2 M=mathrmrad(mathrmrad M)$.



      All the modules or algebras here are finite dimensional over some field $k$. A little more properties is assumed but seems not considered. Such as $Q$ is uniserial with length $2$, with composition factors isomorphic to $k$.



      I wonder why this is true. An epimorphism may not send radical onto radical. By definition, the quotients are semisimple. It seems to say that all the simple submodules to appear in $mathrmrad Q/ mathrmrad^2$ must appear in $mathrmrad P/ mathrmrad^2 P$. But I have no idea how to show this.






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      edited Sep 4 at 13:13

























      asked Sep 4 at 12:57









      Kirby Lee

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