# How can I get the surjective homomorphism map from the finitely generated free algebra to a finitely generated algebra?

We know that every f.g. algebra is isomorphic to quotient of f.g. free algebra. How can I get the surjective homomorphism map from the finitely generated free algebra A=K to a finitely generated algebra B ? (e.g., take B=K+K (direct sum) with generators (1,0),(0,1) )

• the finitely generated free algebra A=K<x1,...,xn>
– Ramy
Dec 7 '14 at 20:39

Let $X=\{x_1,x_2,\ldots,x_n\}$ be a generating set for the finitely generated algebra $B$. The map is given by extending the function $X\to B$ sending $x_i\mapsto x_i$ to a homomorphism from the free algebra generated by $X$. A unique such homomorphism exists by definition of the free algebra, and it is surjective by virtue of the fact that $X$ is a generating set.