Please give me EXAMPLE and explain about:
- Commutative ring and uncommutative ring which doesn't have unit element
- Ring which has different unit element with its subring's unit element
thank you
$\displaystyle n\mathbb{Z}, \ n > 1,$ for the commutative case and $\displaystyle \left \{ \begin{pmatrix} a & b \\ 0 & 0 \end{pmatrix}: \ a,b \in \mathbb{R} \right \}$ for the noncommutative case.
$\displaystyle M_2(\mathbb{R})$ and the subring $\displaystyle \left \{\begin{pmatrix}a & 0 \\ 0 & 0 \end{pmatrix}: \ a \in \mathbb{R} \right \}.$- Ring which has different unit element with its subring's unit element
thank you