In
mathematics, particularly in
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...
, a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
''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
In mathematics, more specifically in the field of ring theory, a ring has the invariant basis number (IBN) property if all finitely generated free left modules over ''R'' have a well-defined rank. In the case of fields, the IBN property becomes ...
(IBN) property. Namely, any
nontrivial[A trivial ring is stably finite but doesn't have IBN.] stably finite ring has IBN.
Commutative rings,
noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
s and
artinian ring In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are n ...
s are stably finite.
Subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
s of stably finite rings and
matrix ring
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ...
s over stably finite rings are stably finite. A ring satisfying
Klein's nilpotence condition is stably finite.
References
* P.M. Cohn (2003). Basic Algebra, Springer.
Ring theory
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