Let $V$ be a finite dimensional vector space and $I$ be an ideal. The tensor algebra of $V$ is defined by $$T(V)= \bigoplus_n V^{\otimes n}$$
Is it true that $T(V)/I $ is isomorphic to $$\bigoplus_n \left(V^{\otimes n}/I \right)?$$
It seems like the ideal doesn't matter if we put the direct sum outside the quotient.