$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Spec{Spec}$Let $A$ be a commutative ring not necessarily with unit and $\mathbb{Z}_2 =\{0,1\}$ be the field of two elements. I am looking for a paper describing the structure of the set of *ring* homomorphisms $\Hom(A,\mathbb{Z}_2)$.
So this is a specific subset of the set of all homomorphisms $A\to\mathbb{Z}_2$ as abelian groups.

If $\phi:A \to \mathbb{Z}_2$ is a non-zero ring homomorphism, then its kernel is a two-sided maximal ideal of $A$ having index $2$ as an abelian subgroup of $A$. And conversely, every such ideal $K$ determines a unique ring homomorphism $A \to A/K = \mathbb{Z}_2$.

Notice that $K$ can be regarded as a certain point of the spectrum $\Spec(A)$. Hence $\Hom(A,\mathbb{Z}_2)$ is a subset of $\Spec(A)$.

In the paper

- Joseph A. Gallian, James Van Buskirk. The number of homomorphisms from $\mathbb{Z}_m$ into $\mathbb{Z}_n$, Amer. Math. Monthly 91 (1984), no. 3, 196-197, doi:10.1080/00029890.1984.11971375

it is proved that the number of ring homomorphisms $\Hom(\mathbb{Z}_m,\mathbb{Z}_n)$ equals $2^{\omega(n)-\omega(n/\gcd(m,n))}$, where $\omega(a)$ is the number of distinct prime factors of $a$. See also this question on MathOverflow.

In particular,

- $\omega(1)=0$,
- $\omega(2)=1$,
- $\Hom(\mathbb{Z}_{2m},\mathbb{Z}_2)$ consists of two homomorphisms(zero and $x\mapsto x~\mathrm{mod}~2$),
- $\Hom(\mathbb{Z}_{2m+1},\mathbb{Z}_2)$ consists of a unique zero homomorphism.

Could you please provide any references to the papers studying the set of ring homomorphisms $\Hom(A,\mathbb{Z}_2)$.

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