On the Steiner number of zero-divisor graphs of finite commutative rings

Let R be a finite commutative ring with unity. The zero-divisor graph Γ(R) is constructed on the set Z(R)*=Z(R)\{0}, where two distinct elements are connected by an edge if and only if their product equals zero. In this work, we study the Steiner number of Γ(R), particularly for cases where R=ℤm, a reduced ring, or a finite direct product of rings of the type Zm. We present a closed-form expression for the Steiner number based on the algebraic structure of Zm, employing equi-annihilator class analysis and the H-join operation. A notable finding is that no odd integer can take the form m−ϕ(m)−1+(h−∑i=1h‍pi), where m=p1a1p2a2…phah; this expression represents the Steiner number of Γ(Zm). This result reveals a structural constraint on the Steiner number and encourages further exploration of its parity behavior in different classes of commutative rings.