ADDED: I hadn't heard previously about the recent death of Harsh Pittie. I was somewhat acquainted with him when we were both at NYU-Courant decades ago and recall hearing some of his lectures on topology of Lie groups. His paper from that period was grounded in topology and K-theory, but Steinberg's follow-up (in his typical concise style) rounded out the discussion of representation rings in a more algebraic framework. Moreover, Steinberg exhibits an explicit basis for $R(T)$ as a free $R(G)$-module in the crucial case where $G$ is a semisimple simply connected compact Lie group and $T$ any maximal torus. In particular, the rank here is the order of the Weyl group $W$. (He also observes that the same ideas work for algebraic groups over any algebraically closed field.)
Though I've never worked through the details of Steinberg's paper carefully, the underlying idea can be observed (in an oversimplified way) in the rank 1 case. Denoting the weight lattice (character group of $T$ in additive notation) by $X$, the respective representation rings look like $\mathbb{Z}[X]$ and $\mathbb{Z}[X]^W$. Then Steinberg's basis elements, one for each element $w \in W$, are defined by applying $w^{-1}$ to a product of symbols (in my notation $e^\lambda$) with $\lambda$ running over suitable fundamental weights. In rank 1, the basis just consists of $e^0, e^{-\rho}$.
