Let $k$ be a field and consider the (unital and associative) free algebra on $k$ with two generators ($x$ and $y$), $A= k < x,y >$. I have two basic questions concerning this algebra:
1) If I define a function $f$ on $A$ by putting $f(x)=a$ and $f(y)=b$ (with $a,b$ in some other algebra over $k$), is it true that $f$ always extends to an algebra homomorphism $\phi$ of $A$ with $\phi(x)=a$ and $\phi(y)=b$, i.e., if I chose the images of the generators do I always have the certainty that I'm going to produce an algebra homomorphism in that way?
2) If I have an algebra homomorphism on $A$ and want to define an algebra homomorphism on the quotient algebra $B=A / (x^2-yx+x)$ (for example), if I only check that $\phi(x^2-yx+x)=0$, does it follow that $\phi$ induces an algebra homomorphism on $B$?
Thank you in advance!
Yes. This is the definition of the free algebra, and of the quotient algebra. Explicitly, one sends $\sum_w k_w x^w$ to $\sum_w k_w a^w$, where $w$ runs over words in the free monoid on two generators. (e.g. $xy^2\mapsto ab^2$, etc.) That's the only possible choice, and it does give a homomorphism because the free algebra has no relations. The second question also has an affirmative answer because, to check that a homomorphism out of $A$ extends to one out of $B$, you just have to check that all the cosets of $x^2-yx-x$ have a well defined image.