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Math Help - Rings

  1. #1
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    Rings

    Consider the set of matrices S={[0 r, 0 0] | r is a real number} with normally defined addition and multiplication of matrices.

    a) Show that S is a subring of M(R).
    b) Show that S is not a field.

    I know that for S to be a subring of M(R)is has to be closed under addition, but not multiplication. What does the normally defined addition and multiplication statement mean?
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  2. #2
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    Quote Originally Posted by empressA View Post
    Consider the set of matrices S={[0 r, 0 0] | r is a real number} with normally defined addition and multiplication of matrices.

    a) Show that S is a subring of M(R).
    b) Show that S is not a field.

    I know that for S to be a subring of M(R)is has to be closed under addition, but not multiplication.
    No, a subring must also be closed under multiplication. Saying that it is a ring rather than a field means that there may be elements other than 0 which do not have multiplicative inverses.

    What does the normally defined addition and multiplication statement mean?
    The nddition addition and multiplication of matrices, of course.

    \begin{bmatrix}0 & r \\ 0 & 0\end{bmatrix}+ \begin{bmatrix}0 & s \\ 0 & 0\end{bmatrix}= \begin{bmatrix}0 & r+s \\ 0 & 0\end{bmatrix}

    \begin{bmatrix}0 & r \\ 0 & 0\end{bmatrix}\begin{bmatrix}0 & s \\ 0 & 0\end{bmatrix} = ?
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  3. #3
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    ok...but when i multiply those matrices i end up with [0 0, 0 0]...would that still make it closed under multiplication..i know that it satisifes all the other requirements for a ring but am not sure if its closed under addition. Thats what I meant in the beginning I worded it wrong
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