Idealized Extensions of Artin Algebras and Finitistic Dimensions

Abstract

The article compares the finitistic dimensions of two algebras involved in an idealized extension of Artin algebras under certain conditions. Precisely, we prove that, for the extension of Artin algebras B ⊂ A with the same identity, under the assumption that the Jacobson radical rad(B) of B is an ideal of A and that the quotient algebra A/rad(B) has Loewy length 2, if the global dimension of A is at most 4, then the finitistic dimension of B is finite.

Key Words:

• Finitistic dimension
• Idealized extension
• Left idealized extension

2010 Mathematics Subject Classification:

• 16E10
• 16S70

ACKNOWLEDGMENT

The author is greatly indebted to his supervisor Professor Changchang Xi for his guidance and many helpful suggestions. The author also thanks the referee for helpful comments.