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Math Help - M_n Matrix ring

  1. #1
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    M_n Matrix ring

    Let M_n(A) be the ring of matrixes with elements from the ring A. T_n(A) \subset M_n(A) the set of matrixes in which all elements below the diagonal are 0. T'_n(A) \subset T_n(A) the set of matrixes with the elements in the principal diagonal are 0. D_n(A) the set of diagonal matrixes.

    Prove that,

    T'_n(A) \cong T_n(A)/D_n(A)
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  2. #2
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    Quote Originally Posted by roporte View Post
    Let M_n(A) be the ring of matrixes with elements from the ring A. T_n(A) \subset M_n(A) the set of matrixes in which all elements below the diagonal are 0. T'_n(A) \subset T_n(A) the set of matrixes with the elements in the principal diagonal are 0. D_n(A) the set of diagonal matrixes.

    Prove that,

    T'_n(A) \cong T_n(A)/D_n(A)
    I do it for n=3 it will be clear.

    Define the mapping \begin{bmatrix} a & b & c \\ 0 & d & e \\ 0 & 0 & f \end{bmatrix} \mapsto \begin{bmatrix} 0 & b & c \\ 0 & 0 & e \\ 0 & 0 & 0 \end{bmatrix}

    Then the kernel of this mapping ( \theta: T_n(A) \to T'_n(A)) is D_n and range is T'_n(A).

    By the fundamental isomorphism theorem,
    T'_n(A) \simeq T_n(A)/D_n(A)
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