This is the first time I post a question on MO, so I'm shy a liite bit. Can you give a "nontrivial" example of a finite dimensional hereditary algebra which is quotient of an infinite dimensional algebra ? By "nontrivial" I mean not by killing loops in the path algebra of some quiver, for example $k[X_1] \times\ldots \times k[X_n]/((X_1)\times\ldots\times (X_n))$.
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If you use generators and relations, then any algebra is a quotient of an infinitedimensional algebra, i.e., a quotient of the free associative algebra corresponding to the generators you pick. 

