# Thread: Ring with unity 1 that has subring with unity 1'

1. ## 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!

2. 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 $1_S = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} \neq 1_R.$