×

Finitistic dimension of Artinian rings with vanishing radical cube. (English) Zbl 0707.16002

For left Artinian rings \(\Lambda\) with the property that the cube of the Jacobson radical \(J\) is zero, we confirm the longstanding conjecture that the supremum, \(\text{fin dim}(\Lambda)\), of the projective dimensions of those finitely generated left \(\Lambda\)-modules which have finite projective dimension, is finite. In fact, we prove the conjecture under various relaxed finiteness prerequisites for \(J^ 3\). A major portion of our argument is algorithmic in nature, permitting the computation of tight upper bounds for \(\text{fin dim}(\Lambda)\), based on information about the composition factors of the indecomposable projective left ideals of \(\Lambda\). The roughest form of the bounds obtained in case \(J^ 3=0\) (or more generally, for the case where all simple composition factors of the left \(\Lambda\)-module \(J^ 3\) have finite projective dimension) is as follows: \(\text{fin dim}(\Lambda)\leq 2m+d+1\), where \(m\) is the number of isomorphism types of simple left \(\Lambda\)-modules of infinite projective dimension and \(d\) is the supremum of the projective dimensions of those simple modules having finite projective dimension. An application of the results yields further classes of rings for which the Nakayama Conjecture is known to hold.

MSC:

16E10 Homological dimension in associative algebras
16N20 Jacobson radical, quasimultiplication
16P20 Artinian rings and modules (associative rings and algebras)
16D25 Ideals in associative algebras
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Auslander, M., Buchsbaum, D.: Homological dimension in noetherian rings II. Trans. Am. Math. Soc.88, 194–206 (1958) · Zbl 0082.03402
[2] Bass, H.: Finitistic dimension and a homological generalization of semiprimary rings. Trans. Am. Math. Soc.95, 466–488 (1960) · Zbl 0094.02201
[3] Bass, H.: Injective dimension in noetherian rings. Trans. Am. Math. Soc.102, 18–29 (1962) · Zbl 0126.06503
[4] Fuller, K.R., Zimmermann-Huisgen, B.: On the Generalized Nakayama Conjecture and the Cartan determinant problem. Trans. Am. Math. Soc.294, 679–691 (1986) · Zbl 0593.16010
[5] Gordon, R., Green, E.L.: Modules with cores and amalgamations of indecomposable modules. Mem. Am. Math. Soc.187, 1–145 (1977) · Zbl 0363.16015
[6] Green, E.L., Kirkman, E., Kuzmanovich, J.: Finitistic dimension of finite dimensional monomial algebras. J. Algebra136, 37–50 (1991) · Zbl 0727.16003
[7] Harada, M.: QF-3 and semi-primary PP-rings I. Osaka J. Math.2, 357–368 (1965) · Zbl 0166.30404
[8] Mochizuki, H.: Finitistic global dimension for rings. Pac. J. Math.15, 249–258 (1965) · Zbl 0154.28603
[9] Mueller, B.J.: Dominant dimension of semi-primary rings. J. Reine Angew. Math.232, 173–179 (1968) · Zbl 0165.35303
[10] Raynaud, M., Gruson, L.: Critères de platitude et de projectivé. Invent. Math.13, 1–89 (1971) · Zbl 0227.14010
[11] Small, L.W.: A change of rings theorem. Proc. Am. Math. Soc.19, 662–666 (1968) · Zbl 0162.33902
[12] Tachikawa, H.: Quasi-Frobenius rings and their generalizations. (Lect. Notes Math., vol. 351), Berlin Heidelberg New York: Springer 1973 · Zbl 0271.16004
[13] Zimmermann-Huisgen, B.: Bounds on finitistic and global dimension of Artinian rings with vanishing radical cube (preprint) · Zbl 0808.16009
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.