Definition of the intersection number

Question

Let $M, A, B$ be closed, connected, smooth and oriented (i.e. $\mathbb{Z}$-oriented) manifolds of dimensions $m, a, b$ respectively. Moreover $A,B\subseteq M$ are submanifolds of $M$ of complementary dimensions, i.e. $a+b=m$. Suppose $A$ and $B$ are transverse in $M$. Fix the orientation of $A, B$ and $M$. Then, at any $p\in A\cap B$, we have two bases of $T_pM$, namely $\{T_pA,T_pB\}$ and $\{T_pM\}$ (we choose the local coordinates according to the chosen orientations). Let $\eta_p$ be the sign of transition matrix from basis $\{T_pA,T_pB\}$ of $T_pM$, to $\{T_pM\}$. We define then the $\textbf{intersection number}$ of $A$ and $B$ in $M$ to be $$ \Sigma_{p\in A\cap B}\eta_p. $$ My question is about the definition of the intersection number in case $A,B$ and $M$ are $R$-orientable for some ring $R$ - not necessarily $\mathbb{Z}$. Can one extend this definition to this more general case? How should one define then the $R$-orientation preserving choice of bases of $TM$, $TA$ and $TB$? Is such a definition even possible, or reasonable?