A ring is left hereditary if every submodule of a left projective module is projective. Equivalently, every left module has a projective resolution of length at most one. The standard projective resolution of a quiver representation, used also for arbitrary infinite-dimensional modules, proves that path algebras are left hereditary.
The standard projective resolution of a quiver representation has length one, even for quivers with oriented cycles. Thus every path algebra is a hereditary ring, and higher extension groups vanish. Applying a long exact sequence of Ext groups to gives a surjection , since the next term is zero. This is the hereditary step in the Ringel lemma on bricks.
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A **hereditary ring** is a type of ring in the field of abstract algebra, particularly in ring theory. A ring \( R \) is called hereditary if every finitely generated module over \( R \) is a projective module. This is equivalent to saying that all submodules of finitely generated projective modules are also projective. In simpler terms, projective modules are those that resemble free modules in terms of their structure and properties.