nLab Goldie rank

Definition

Given a left and right Noetherian ring, its Goldie rank is the maximum number of nonzero internal direct summands of its left (or right) ideals. In other words, the Goldie rank of a Noetherian ring $R$ is $n$ if $R$ contains an internal direct sum of $n$ non-zero? submodules but no internal direct sum of $n+1$ nonzero submodules.

The same definition may be used for the Goldie rank of “finite rank” modules, where it is also called Goldie dimension and uniform dimension. Here by finite rank one means that the injective hull is a direct sum of a finite direct sum of indecomposables. An alternative definition is this: a finite rank module $M$ has Goldie rank $n$ if the injective hull $E(M)$ is a direct sum of n uniform submodules.

Properties

• Goldie rank is preserved under Ore localization at any Ore set of regular elements.
• If the ring is prime, then any Ore extension has the same Goldie rank.

Literature

A textbook account is in

• K. R. Goodearl, R. B. Warfield, Jr., An introduction to noncommutative Noetherian rings,

  London Math. Soc. Student Texts 16, Cambridge Univ. Press 1989

• J. C. McConnell, J. C. Robson, Noncommutative Noetherian rings, Wiley 1987.

• Anthony Joseph, Lance W. Small, An additivity principle for Goldie rank, Israel J. Math. 31 (1978) 105–114 [doi:10.1007/BF02760541]

• Anthony Joseph, Kostant’s problem, Goldie rank and the Gelfand-Kirillov conjecture, Invent. Math. 56 (1980) 191–213

• W. Borho, On the Joseph-Small additivity principle for Goldie ranks, Compositio Math. 47 (1982) 3–29

For the universal enveloping algebras

• Anthony Joseph, Goldie rank in the enveloping algebra of a semisimple Lie algebra, I, J. Algebra 65 2 (1980) 269–283 [doi:10.1016/0021-8693(80)90217-3]; Goldie rank in the enveloping algebra of a semisimple Lie algebra II, J. Algebra 65 (1980) 284–306
• Jonathan Brundan, Mœglin’s theorem and Goldie rank polynomials in Cartan type A, Compositio Math. 147 (2011) 1741–1771 doi
• K. R. Goodearl, E. S. Letzter, Prime factor algebras of the coordinate ring of quantum matrices, Proc. Amer. Math. Soc. 121 (1994) 1017–1025 pdf

category: algebra