I have some confusion that i would like to clarify. The product in cohomology is not the cup product $\smile$ but it is another product $*$ that is constructed from cup product. Indeed wrinte $H^\bullet(X;R) = \bigoplus_{k\in\mathbb{N}} H^k(X; R)$ then an element $x$ in $H^\bullet(X;R)$ is of the form $x=x_0+x_1+x_2+\cdots $ where each $x_i\in H^i(X;R)$ hence the product of two elements $x$ and $y$ in $H^*(X;R)$ is a mutliplication $*$ defined via cup product as follows (we take an example $x=x_1+x_2$ and $y=y_2+y_3$)
$x*y=(x_1+x_2)*(y_2+y_3)=x_1\smile y_2 + x_1\smile y_3+ x_2\smile y_2+x_2\smile y_3$ is this correct?