Well-defindness of a map involving tensor product.

Question

Let $k$ be a field. The identity element of any $k$-algebra $\Gamma$ gives a structure map $I\colon k\to \Gamma$; its cokernel $\Gamma/I(k) = \Gamma /(k.1_{\Gamma})$ will be denoted $\overline{\Gamma}$.

I am having trouble to see that $F$ is well-defined. Can someone give me a hint on how to do this? Thanks!

Answer 1

Let $\bar a = a + k1_A$, and extend the notation to the other algebras in play. The map $\bar A \to \overline{A\otimes B}$ given by $\bar a \mapsto \overline{a \otimes 1}$ is well-defined because if $\bar a_1 = \bar a_2$ then $a_1$ and $a_2$ differ by a multiple of $1_A$, so $a_1 \otimes 1_B$ and $a_2\otimes 1_B$ differ by a multiple of $1_A\otimes 1_B$, which is zero in $\overline{A\otimes B}$.