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Math Help - Ring with unity 1 that has subring with unity 1'

  1. #1
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    Ring with unity 1 that has subring with unity 1'

    Hi, let me just say that I'm not in a class or anything, I'm just working through an old book and trying to understand.

    Problem: Give an example of a ring with unity 1 that has a subring with unity 1' \not= 1.

    So my ring must be not commutative or be such that not every nonzero element has a multiplicative inverse, since in a field "the unity element of a subfield must be the unity of the whole field." But I can't find an example. Thanks in advance!
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  2. #2
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    let R be the ring of all 2 \times 2 matrices with integer entries. Let S=\left \{ \begin{pmatrix} a & 0 \\ 0 & 0 \end{pmatrix}: \ a \in \mathbb{Z} \right \}. then S is a subring of R and 1_S = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \neq 1_R.
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