# properties of the ordinary quiver

Let $k$ be a field and $A$ be a finite-dimensional algebra over $k$. Denote by $Q_{A}$ the ordinary quiver (http://planetmath.org/OrdinaryQuiverOfAnAlgebra) of $A$.

Theorem. The following statements hold:

1. 1.

If $A$ is basic and connected, then $Q_{A}$ is a connected quiver.

2. 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. 3.

If $A$ is basic and connected, then $A$ is isomorphic to $kQ_{A}/I$ for some (not necessarily unique) admissible ideal (http://planetmath.org/AdmissibleIdealsBoundQuiverAndItsAlgebra) $I$.

For proofs please see [1, Chapter II.3].

## References

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