quotient module with dimensional 1

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If $V=bigoplus V_i$ is an n-dimensional $R$-module and $W$ is a hyperplane "submodule of V" whose coordinates sum equal to zero (thus it is of $ n-1$ dimension). If the quotient space $V/W$ is also an $R$-module; thus it's of $ 1$-dimension. I want to show that bais of the quotient module is $v_i+W$ no matter what $v_iin V_i$ is. I mean even $v_i+w=v_j+W.$ Can you please provide me with a hint?




linear-algebra group-theory modules representation-theory




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  • $v_i+W=v_j+Wiff v_i-v_jin W$
    – Lord Shark the Unknown
    Aug 1 at 17:49










  • Does it mean that the only coset of the quotient we have is the $W$?
    – Mal JA
    Aug 1 at 17:54










  • What is $R$? You mean $mathbbR$?
    – Batominovski
    Aug 1 at 19:02










  • No it is a ring.
    – Mal JA
    Aug 1 at 21:42














up vote
0
down vote

favorite












If $V=bigoplus V_i$ is an n-dimensional $R$-module and $W$ is a hyperplane "submodule of V" whose coordinates sum equal to zero (thus it is of $ n-1$ dimension). If the quotient space $V/W$ is also an $R$-module; thus it's of $ 1$-dimension. I want to show that bais of the quotient module is $v_i+W$ no matter what $v_iin V_i$ is. I mean even $v_i+w=v_j+W.$ Can you please provide me with a hint?




linear-algebra group-theory modules representation-theory




share|cite|improve this question



















  • $v_i+W=v_j+Wiff v_i-v_jin W$
    – Lord Shark the Unknown
    Aug 1 at 17:49










  • Does it mean that the only coset of the quotient we have is the $W$?
    – Mal JA
    Aug 1 at 17:54










  • What is $R$? You mean $mathbbR$?
    – Batominovski
    Aug 1 at 19:02










  • No it is a ring.
    – Mal JA
    Aug 1 at 21:42












up vote
0
down vote

favorite









up vote
0
down vote

favorite











If $V=bigoplus V_i$ is an n-dimensional $R$-module and $W$ is a hyperplane "submodule of V" whose coordinates sum equal to zero (thus it is of $ n-1$ dimension). If the quotient space $V/W$ is also an $R$-module; thus it's of $ 1$-dimension. I want to show that bais of the quotient module is $v_i+W$ no matter what $v_iin V_i$ is. I mean even $v_i+w=v_j+W.$ Can you please provide me with a hint?




linear-algebra group-theory modules representation-theory




share|cite|improve this question











If $V=bigoplus V_i$ is an n-dimensional $R$-module and $W$ is a hyperplane "submodule of V" whose coordinates sum equal to zero (thus it is of $ n-1$ dimension). If the quotient space $V/W$ is also an $R$-module; thus it's of $ 1$-dimension. I want to show that bais of the quotient module is $v_i+W$ no matter what $v_iin V_i$ is. I mean even $v_i+w=v_j+W.$ Can you please provide me with a hint?





linear-algebra group-theory modules representation-theory





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Aug 1 at 17:48









Mal JA

333




333











  • $v_i+W=v_j+Wiff v_i-v_jin W$
    – Lord Shark the Unknown
    Aug 1 at 17:49










  • Does it mean that the only coset of the quotient we have is the $W$?
    – Mal JA
    Aug 1 at 17:54










  • What is $R$? You mean $mathbbR$?
    – Batominovski
    Aug 1 at 19:02










  • No it is a ring.
    – Mal JA
    Aug 1 at 21:42
















  • $v_i+W=v_j+Wiff v_i-v_jin W$
    – Lord Shark the Unknown
    Aug 1 at 17:49










  • Does it mean that the only coset of the quotient we have is the $W$?
    – Mal JA
    Aug 1 at 17:54










  • What is $R$? You mean $mathbbR$?
    – Batominovski
    Aug 1 at 19:02










  • No it is a ring.
    – Mal JA
    Aug 1 at 21:42















$v_i+W=v_j+Wiff v_i-v_jin W$
– Lord Shark the Unknown
Aug 1 at 17:49




$v_i+W=v_j+Wiff v_i-v_jin W$
– Lord Shark the Unknown
Aug 1 at 17:49












Does it mean that the only coset of the quotient we have is the $W$?
– Mal JA
Aug 1 at 17:54




Does it mean that the only coset of the quotient we have is the $W$?
– Mal JA
Aug 1 at 17:54












What is $R$? You mean $mathbbR$?
– Batominovski
Aug 1 at 19:02




What is $R$? You mean $mathbbR$?
– Batominovski
Aug 1 at 19:02












No it is a ring.
– Mal JA
Aug 1 at 21:42




No it is a ring.
– Mal JA
Aug 1 at 21:42















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