Units in Upper Triangular Matrix Rings

Detailed Solution

Every element of $R$ has the form $$ A=\begin{pmatrix} a&b\\ 0&c \end{pmatrix}, \qquad a,b,c\in\mathbb Z_2. $$ Since each of $a,b,c$ has two possible values, the ring contains $$ 2^3=8 $$ elements. A square matrix over a field is invertible if and only if its determinant is nonzero. Here, $\mathbb Z_2$ is a field, and $$ \det(A)=ac. $$ Therefore, $A$ is invertible precisely when $$ ac=1. $$ In $\mathbb Z_2$, this requires $$ a=1,\qquad c=1. $$ The entry $b$ can be either $0$ or $1$. Thus the only units are $$ \begin{pmatrix}1&0\\0&1\end{pmatrix} \quad\text{and}\quad \begin{pmatrix}1&1\\0&1\end{pmatrix}. $$ Consequently, the ring has exactly $$ 2 $$ units. Thus the correct option is $\boxed{\mathrm{B}}$.