properties of the ordinary quiver
Let $k$ be a field and $A$ be a finitedimensional algebra over $k$. Denote by ${Q}_{A}$ the ordinary quiver (http://planetmath.org/OrdinaryQuiverOfAnAlgebra) of $A$.
Theorem. The following statements hold:

1.
If $A$ is basic and connected, then ${Q}_{A}$ is a connected quiver.

2.
If $Q$ is a finite quiver and $I$ is an admissible ideal (http://planetmath.org/AdmissibleIdealsBoundQuiverAndItsAlgebra) in $kQ$ and $A=kQ/I$, then ${Q}_{A}$ and $Q$ are isomorphic^{}.

3.
If $A$ is basic and connected, then $A$ is isomorphic to $k{Q}_{A}/I$ for some (not necessarily unique) admissible ideal (http://planetmath.org/AdmissibleIdealsBoundQuiverAndItsAlgebra) $I$.
For proofs please see [1, Chapter II.3].
References
 1 I. Assem, D. Simson, A. Skowronski, Elements of the Representation Theory of Associative Algebras, vol 1., Cambridge University Press 2006, 2007
Title  properties of the ordinary quiver 

Canonical name  PropertiesOfTheOrdinaryQuiver 
Date of creation  20130322 19:17:44 
Last modified on  20130322 19:17:44 
Owner  joking (16130) 
Last modified by  joking (16130) 
Numerical id  4 
Author  joking (16130) 
Entry type  Theorem 
Classification  msc 16S99 
Classification  msc 20C99 
Classification  msc 13B99 