properties of admissible ideals

Let Q be a quiver, k a field and I an admissible ideal (see parent object) in the path algebraPlanetmathPlanetmath kQ. The following propositionsPlanetmathPlanetmath and proofs are taken from [1].

Proposition 1. If Q is finite, then kQ/I is finite dimensional algebra.

Proof. Let RQ be the arrow ideal in kQ. Since RQmI for some m, then we have a surjectivePlanetmathPlanetmath algebra homomorphism kQ/RQmkQ/I. Thus, it is enough to show, that kQ/RQm is finite dimensional. But since Q is a finite quiver, then there is finitely many paths of length at most m. It is easy to see, that these paths form a basis of kQ/RQm as vector space over k. This completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Proposition 2. If Q is finite, then I is a finitely generatedMathworldPlanetmathPlanetmath ideal.

Proof. Consider the short exact sequenceMathworldPlanetmathPlanetmath


of kQ modules. It is well known that in such sequences the middle term is finitely generated if the end terms are. Of course RQm is finitely generated, because Q is finite so there is finite number of paths of length m.

On the other hand I/RQm is an ideal in kQ/RQm, which is finite dimensional by proposition 1. Thus I/RQm is a finite dimensional vector space over k. But then it is finitely generated kQ module (see this entry ( for more details), which completes the proof.

Proposition 3. If Q is finite, then there exists a finite setMathworldPlanetmath of relationsMathworldPlanetmathPlanetmathPlanetmath ( {ρ1,,ρm} such that I is generated by them.

Proof. By proposition 2 there is a finite set of generatorsPlanetmathPlanetmathPlanetmath {a1,,an} of I. Generally the don’t have to be relations. On the other hand, if ex denotes the stationary path in xQ0, then it can be easily checked, that every element of the form exaiey is either zero or a relation. Also, note that


Since Q is finite, then this completes the proof.


  • 1 I. Assem, D. Simson, A. SkowroÃski, Elements of the Representation Theory of Associative Algebras, vol 1., Cambridge University Press 2006, 2007
Title properties of admissible ideals
Canonical name PropertiesOfAdmissibleIdeals
Date of creation 2013-03-22 19:16:48
Last modified on 2013-03-22 19:16:48
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Theorem
Classification msc 14L24