Let $k$ be a field and $V$ a $k$-vector space. Denote the inclusion of $k$ in $k \oplus V$ by $m$. A book that I'm consulting (Ramanan's Global Calculus) asks the reader to prove that the quotient of the tensor algebra of $k \oplus V$ by the 2-sided ideal generated by $1-m(1)$ is canonically isomorphic to the tensor algebra of $V$. I'm not quite sure how to do this.
There is a natural map from $k \oplus V \to V$. This induces a map between the respective tensor algebras, but the kernel of this map seems to be generated by $m(1)$ and not $1 - m(1)$. I'm not quite sure how to construct a map whose kernel is as required.
Hint: if $T(V)$ is the tensor algebra on $V$, define $f: k \oplus V \to T(V)$ by $f(x, v) = x + v$ (where I am identifying $k$ and $v$ with their images in $T(V)$ expressed as the sum $k \oplus V \oplus V^{{\otimes}2} \oplus \ldots$). Then $f$ induces an algebra homomorphism $T(k \oplus V) \to T(V)$ by the universal property of the tensor algebra.