Let $R$ be a ring. Given an exact sequence of $R$-modules $$K\longrightarrow N\longrightarrow M\longrightarrow0$$ deduce $$T(N)\otimes K\otimes T(N)\longrightarrow T(N)\longrightarrow T(M)\longrightarrow0$$ is exact.
Here we define for any $R$-module $M$, $T(M):=R\oplus M\oplus (M\otimes_RM)\oplus(M\otimes_RM\otimes_RM)\oplus\dots$
Is it possible to use right exactness of tensor product to prove this ?