I am interested in computing the cohomology ring $H^*(\mathbb{R}P^3 \# \mathbb{R}P^3; \mathbb{Z}_2)$. Here # is the connected sum. Using a suggestion here on my earlier post, I computed the additive structure as $H^i(\mathbb{R}P^3 \# \mathbb{R}P^3; \mathbb{Z}_2)=\begin{cases} \mathbb{Z}_2 & \mbox{if } i=0 \\ \mathbb{Z}_2 \oplus \mathbb{Z}_2 & \mbox{if } i=1 \\ \mathbb{Z}_2 \oplus \mathbb{Z}_2 & \mbox{if } i=2 \\ \mathbb{Z}_2 & \mbox{if } i=3 \\ \end{cases}$
But I am having hard time in computing the ring structure. I actually want to compute $H^*(\mathbb{R}P^n \# \mathbb{R}P^n; \mathbb{Z}_2)$ for odd $n$ and think that the case $n=3$ should help me get the general case. Ths case $n=1$ is trivial as $\mathbb{R}P^1 \# \mathbb{R}P^1= \mathbb{S}^1$.