Characteristic of a Quotient Ring

Detailed Solution

In the quotient ring $$ R=\mathbb Z_2[x]/(x^2), $$ the polynomial $x^2$ is identified with $0$. Thus, every element of $R$ has a unique representative of the form $$ a+bx,\qquad a,b\in\mathbb Z_2. $$ Consequently, the four elements of $R$ are $$ R=\{0,1,x,1+x\}. $$ Hence $|R|=4$. The characteristic of a ring with unity is the least positive integer $n$ such that $$ n\cdot 1_R=0. $$ Since $\mathbb Z_2$ has characteristic $2$, we have $$ 1_R+1_R=0. $$ Therefore, $$ 2\cdot 1_R=0, $$ while $1_R\neq0$. Hence the characteristic of $R$ is exactly $$ \operatorname{char}(R)=2. $$ Thus option B is correct. We also verify the remaining options. Option A is incorrect because the characteristic of a ring need not equal its number of elements. Here, $|R|=4$, but $\operatorname{char}(R)=2$. Option C is incorrect because $x\neq0$ in the quotient ring, while $$ x^2=0. $$ Therefore, $R$ contains a nonzero nilpotent element. A field has no nonzero nilpotent elements, so $R$ cannot be a field and is not isomorphic to $\mathbb F_4$. Option D is incorrect for the same reason: $x$ is a nonzero nilpotent element because $x^2=0$. Thus the correct option is $\boxed{\mathrm{B}}$.