Stably finite ring (original) (raw)
In mathematics, particularly in abstract algebra, a ring R is said to be stably finite (or weakly finite) if, for all square matrices A and B of the same size with entries in R, AB = 1 implies BA = 1. This is a stronger property for a ring than having the invariant basis number (IBN) property. Namely, any nontrivial stably finite ring has IBN. Commutative rings, noetherian rings and artinian rings are stably finite. Subrings of stably finite rings and matrix rings over stably finite rings are stably finite. A ring satisfying is stably finite.
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| dbo:abstract | In mathematics, particularly in abstract algebra, a ring R is said to be stably finite (or weakly finite) if, for all square matrices A and B of the same size with entries in R, AB = 1 implies BA = 1. This is a stronger property for a ring than having the invariant basis number (IBN) property. Namely, any nontrivial stably finite ring has IBN. Commutative rings, noetherian rings and artinian rings are stably finite. Subrings of stably finite rings and matrix rings over stably finite rings are stably finite. A ring satisfying is stably finite. (en) Inom matematiken är en ring stabilt ändlig (eller svagt ändlig) om för alla kvadratiska matriser A, B av samma storlek över R implicerar AB = 1 att BA = 1. (sv) |
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| rdfs:comment | In mathematics, particularly in abstract algebra, a ring R is said to be stably finite (or weakly finite) if, for all square matrices A and B of the same size with entries in R, AB = 1 implies BA = 1. This is a stronger property for a ring than having the invariant basis number (IBN) property. Namely, any nontrivial stably finite ring has IBN. Commutative rings, noetherian rings and artinian rings are stably finite. Subrings of stably finite rings and matrix rings over stably finite rings are stably finite. A ring satisfying is stably finite. (en) Inom matematiken är en ring stabilt ändlig (eller svagt ändlig) om för alla kvadratiska matriser A, B av samma storlek över R implicerar AB = 1 att BA = 1. (sv) |
| rdfs:label | Stably finite ring (en) Stabilt ändlig ring (sv) |
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