Just two quick questions. I am to find 1. A unitary ring R with a non-unitary subring S, and 2. A non-unitary ring R with subring S which is itself a unitary ring.
I think I understand the ideas, and here is my attempt:
1. \(\displaystyle R = \mathbb{Z}\) and S = \(\displaystyle 2\mathbb{Z}\). \(\displaystyle 1 \in R, \,\, 1\not{\in} S\)
2. R = {\(\displaystyle A \in M_2(\mathbb{R}) \,\, | \,\,\) det \(\displaystyle (A)=0}\) and \(\displaystyle S =\) { \(\displaystyle [\begin{array} a & 0 \\ 0 & 0 \end{array}] \,\, | \,\, a \in \mathbb{R}\)}... I \(\displaystyle \not{\in} R\), but the matrix \(\displaystyle [\begin{array} 1 & 0 \\ 0 & 0 \end{array}]\) is an identity element of S.
I am still trying to digest all these new terms, so I'm not sure if they're correct. Whether correct or not, are there any other examples of a Non-unitary ring containing a subring that is itself a unitary ring?
Thanks
-Daon
I think I understand the ideas, and here is my attempt:
1. \(\displaystyle R = \mathbb{Z}\) and S = \(\displaystyle 2\mathbb{Z}\). \(\displaystyle 1 \in R, \,\, 1\not{\in} S\)
2. R = {\(\displaystyle A \in M_2(\mathbb{R}) \,\, | \,\,\) det \(\displaystyle (A)=0}\) and \(\displaystyle S =\) { \(\displaystyle [\begin{array} a & 0 \\ 0 & 0 \end{array}] \,\, | \,\, a \in \mathbb{R}\)}... I \(\displaystyle \not{\in} R\), but the matrix \(\displaystyle [\begin{array} 1 & 0 \\ 0 & 0 \end{array}]\) is an identity element of S.
I am still trying to digest all these new terms, so I'm not sure if they're correct. Whether correct or not, are there any other examples of a Non-unitary ring containing a subring that is itself a unitary ring?
Thanks
-Daon
