As a graded $\mathbb{Z}$-module, the structure of the group cohomology $H^{\*}(\mathbb{Z}/n\mathbb{Z};\mathbb{Z})$ is extremely well-known. Yet, I am having difficulty finding a reference concerning its cup product structure. I assume this is also well-known, but I would appreciate a reference containing a precise statement of the $\mathbb{Z}$-algebra structure. 



Reference for Ring Structure on Group Cohomology
See Example 3.41, p. 251 of [Hatcher's Algebraic Topology](http://www.math.cornell.edu/~hatcher/AT/ATpage.html). This example computes the cohomology ring of finite Lens spaces. Since the infinite Lens space is a $K(\mathbb{Z}/n,1)$, and is a increasing union of finite Lens spaces, the same computation holds for the infinite Lens spaces. 


**Edit:** My original answer was incomplete, since Example 3.41 computes the ring $H^\*(\mathbb{Z}/n,\mathbb{Z}/n) \cong \mathbb{Z}/n [ \alpha, \beta]/(\alpha^2-k\beta)$ (*where $k=0$ if $n$ is odd, and $k=n/2$ if $n$ is even, see Hatcher's comment below*, and this is an graded-commutative ring with $\alpha$ of degree $1$, and $\beta$ of degree $2$), whereas you want $H^\*(\mathbb{Z}/n,\mathbb{Z})$. 


We have $H^{2i}(\mathbb{Z}/n,\mathbb{Z}) \cong \mathbb{Z}/n$ for $i>0$, $H^0(\mathbb{Z}/n,\mathbb{Z})\cong \mathbb{Z}$, and $H^{2i+1}(\mathbb{Z}/n,\mathbb{Z}) \cong 0$ by universal coefficients (or Poincare duality). Let $\beta\in H^2(\mathbb{Z}/n,\mathbb{Z})$ be a generator, then the map from $H^\*(\mathbb{Z}/n,\mathbb{Z})\to H^\*(\mathbb{Z}/n,\mathbb{Z}/n)$ is a ring isomorphism for even degrees above zero, so we see that $H^\*(\mathbb{Z}/n,\mathbb{Z})\cong \mathbb{Z}[\beta]/(n\beta)$, where $\beta$ has degree $2$ (and is torsion of order $n$). 


