Separable groupoid rings. (English) Zbl 1125.16016
Author’s summary: We show that groupoid rings are separable over their ring of coefficients if and only if the groupoid is finite and the orders of the associated principal groups are invertible in the ring of coefficients. We use this to show that if we are given a finite groupoid, then the associated groupoid ring is semisimple (or hereditary) if and only if the ring of coefficients is semisimple (or hereditary) and the orders of the principal groups are invertible in the ring of coefficients. To this end, we extend parts of the theory of graded rings and modules from the group graded case to the category graded, and, hence, groupoid graded situation. In particular, we show that strongly groupoid graded rings are separable over their principal components if and only if the image of the trace map contains the identity.
Reviewer: Serban Raianu (Carson)
MSC:
16S36 | Ordinary and skew polynomial rings and semigroup rings |
20M25 | Semigroup rings, multiplicative semigroups of rings |
16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |
16W50 | Graded rings and modules (associative rings and algebras) |
20L05 | Groupoids (i.e. small categories in which all morphisms are isomorphisms) |
References:
[1] | Bass H., Algebraic K-Theory (1968) |
[2] | Caenepeel S., Brauer Groups and the Cohomology of Graded Rings (1988) · Zbl 0702.13001 |
[3] | DOI: 10.1016/S0022-4049(96)00181-8 · Zbl 0931.18001 · doi:10.1016/S0022-4049(96)00181-8 |
[4] | DOI: 10.1007/BF01161413 · Zbl 0424.16001 · doi:10.1007/BF01161413 |
[5] | DeMeyer F. M., Separable Algebras over Commutative Rings (1971) · Zbl 0215.36602 |
[6] | DOI: 10.1090/S0002-9947-00-02476-4 · Zbl 0955.16018 · doi:10.1090/S0002-9947-00-02476-4 |
[7] | DOI: 10.1081/AGB-100107943 · Zbl 0993.16013 · doi:10.1081/AGB-100107943 |
[8] | Le Bruyn L., Graded Orders (1988) |
[9] | Năstăsescu C., Graded Ring Theory (1982) |
[10] | DOI: 10.1016/0021-8693(89)90053-7 · Zbl 0673.16026 · doi:10.1016/0021-8693(89)90053-7 |
[11] | Passman D. S., Infinite Group Rings (1971) · Zbl 0247.16004 |
[12] | DOI: 10.1080/00927879008823975 · Zbl 0713.18002 · doi:10.1080/00927879008823975 |
[13] | Reiner I., Maximal Orders. (1975) |
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.