Ring and Subring with different Identities

Question

Is there an example of a ring $S$ with identity $1_S$ containing a non-trivial subring $R$ which itself has an identity $1_R$, but $1_R\neq 1_S$ (or equivalently $1_S\notin R$). I'd also like to know under what conditions the identities have to be equal. I know they must be equal if $S$ has no zero divisors, since for every $r\in R$ we have $(1_S-1_R)r=0$

In other words: Is the category of unital rings with identity-preserving morphisms a full subcategory of the category of rings (where morphism are only required to respect addition and multiplication)?

Answer 1

Hints:

$$S:=\left\{\;\begin{pmatrix}x&y\\z&w\end{pmatrix}\;;\;\;x,y,z,w\in\Bbb R\;\right\}\;,\;R:=\left\{\;\begin{pmatrix}a&0\\0&0\end{pmatrix}\;;\;\;a\in\Bbb R\;\right\}$$

$$1_R=\begin{pmatrix}1&0\\0&0\end{pmatrix}$$

Answer 2

Let $D_4=\langle x,y\mid x^4=y^2=xyxy=1\rangle,$ and consider the group ring $S=\Bbb C[D_4],$ a vector space of dimension $8$ over $\Bbb C$ with canonical standard basis $\{\mathbf{e}_{1},\mathbf{e}_{x},\mathbf{e}_{x^2},\mathbf{e}_{x^3},\mathbf{e}_{y},\mathbf{e}_{xy},\mathbf{e}_{x^2y},\mathbf{e}_{x^3y}\}.$ Multiplication in this vector space is given by $\mathbf{e}_{g}\mathbf{e}_{h}=\mathbf{e}_{gh}$ for any $g,h\in D_4,$ with scalars multiplying as usual, and distributivity as expected. The multiplicative identity element of $S$ is $\mathbf{e}_{1}.$

Since the center of $D_4$ is $\{1,x^2\},$ then $\mathbf{v}_{1}:=\frac12(\mathbf{e}_{1}-\mathbf{e}_{x^2})$ commutes multiplicatively with everything in $S.$ Note further that $\mathbf{v}_{1}^2=\mathbf{v}_{1}$. For each $g\in D_4,$ let $\mathbf{v}_{g}:=\mathbf{v}_{1}\mathbf{e}_{g}.$ It can be shown that $\mathbf{v}_{g}\mathbf{v}_{h}=\mathbf{v}_{gh}$ for all $g,h\in D_4,$ and that $\mathbf{v}_{x^2}=-\mathbf{v}_{1},$ so $$\{\mathbf{v}_{g}\mid g\in D_4\}=\{\pm \mathbf{v}_{1},\pm \mathbf{v}_{x},\pm\mathbf{v}_{y},\pm\mathbf{v}_{x}\mathbf{v}_{y}\},$$ whence $$R:=\{a\mathbf{v}_{1}+b\mathbf{v}_{x}+c\mathbf{v}_{y}+d\mathbf{v}_{x}\mathbf{v}_{y}\mid a,b,c,d\in\Bbb C\}$$ is a subset of $S$ that is closed under the addition and multiplication operations of $R$--a subring of $S$--whose multiplicative identity element is $\mathbf{v}_{1}\ne\mathbf{e}_{1}.$

See Arturo's fine answer here for more on this subject.

Answer 3

Another example is the ring made up of $\begin{pmatrix} a & b\\ c & d\end{pmatrix}$, where $a,b,c,d\in\Bbb{R}$. The identity is $\begin{pmatrix} 1 & 0\\ 0 & 1\end{pmatrix}$.

It has a subring: $\begin{pmatrix} a & a\\ a & a\end{pmatrix}$, where $a\in\Bbb{R}$. The identity of this subring is $\begin{pmatrix} \frac{1}{2} & \frac{1}{2}\\ \frac{1}{2} & \frac{1}{2}\end{pmatrix}$.