Let \(R\) be a commutative ring with unity. This paper studies \(Z(R)\) using homology group, where \(Z(R)\) is the set of zero-divisors in \(R\). In addition, the authors calculate the group \(H_{0}(R)\) in depth, in particular cases, and compute \(H_{1}(\frac{\mathbb{Z}}{p^{r}\mathbb{Z}})\) where \(p\) is a prime and \(r\geq 1\) is an integer. The last section of this paper computes the Euler characteristic \(\chi(R)=\sum_{i=0}^{\infty}(-1)^{i}\text{rk} \;H_{i}(R)\), and the authors show that \(\chi(\frac{\mathbb{Z}}{p^{r}\mathbb{Z}})\) is always \(0\), \(1\) or \(2\) depending on the value of \(r\) relative to the ``the pentagonal'' number \(\frac{m(3m-1)}{2}\) and the related numbers \(\frac{m(3m+1)}{2}\).