# Ring homomorphisms from the commutative ring into $\mathbb{Z}_2$

$$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Spec{Spec}$$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 $$\Hom(A,\mathbb{Z}_2)$$.

• Are you sure Gallian and Buskirk are looking at ring homomorphisms? Or are they counting arbitrary additive homomorphisms? Oct 23, 2022 at 7:22
• Yes, they even wonder that they did not find such a description in the literature before. Please look at their paper here: https://www.d.umn.edu/~jgallian/homs.pdf. They compute both types. An interesting notice is that the number of group homomorphisms $\mathbb{Z}_m\to\mathbb{Z}_n$ equals $gcd(m,n)$ and it is symmetric in $m$ and $n$, while the number of ring homomorphisms is not. Oct 23, 2022 at 7:30
• The Gallian and Buskirk calculation includes homomorphisms which do not preserve 1. In your case, only the zero homomorphism is multiplicative and does not preserve 1: you should exclude this case from your count. Oct 23, 2022 at 7:42
• The only ring homomorphsm $A\to \mathbb Z_2$ that does not preserve 1 is the zero homomorphism. Oct 23, 2022 at 7:51
• Let $J$ be the intersection of all kernels of homomorphisms $A\to\mathbf{F}_2$ and $B=A/J$. Then $B$ is the largest Boolean algebra quotient of $A$. The set you're interested is the spectrum of $B$. It has a natural Hausdorff topology (compact totally disconnected), which coincides with the Zariski topology of $\mathrm{Spec}(B)$. If $A$ is noetherian, then so is $B$, and hence $B$ is finite and $\mathrm{Spec}(B)$ has cardinal $\log_2(|B|)$. (The above concerns unital homomorphisms, but as already mentioned, including non-unital homomorphisms just adds the zero homomorphism.)
– YCor
Oct 23, 2022 at 9:05