On the dimension formula for Goldie's uniform dimension
 
 
Description:  A classical (right) ring of fractions of a ring R is an overring Q(R) of R such that any non-zero divisor of R is invertible in Q(R) and any element of Q(R) is of the form ab^{-1} for a and b elements of R with b a non-zero divisor. Oystein Ore proved in the 30s that a (non-commutative) domain R admits such a ring of fractions, which is then a division ring, if and only if any two non-zero cyclic right ideals of R have a non-zero intersection. A module with the latter condition is termed uniform and a module is said to have finite uniform dimension, whenever it contains a finite direct sum of uniform submodules such that any non-zero submodule has non-zero intersection with this direct sum. In the 50's and 60's Alfred Goldie proved that a ring R has a ring of fractions that is a direct product of matrix rings over division rings if and only if R has no nilpotent ideals, satisfies the ascending chain conditions of right annihilators and has finite uniform dimension as a right module over itself.

After reviewing the lattice theoretical foundations of uniform dimension and its dual, I will discuss, when the classical dimension formula for vector spaces holds for uniform dimensions. In the late 80's Victor Camillo and Julius Zelmanowitz published a paper stating that a ring would be a direct sum of matrix rings over division rings if and only if the dimension formula holds for all modules. However their proof contains a gap, since only modules with finite uniform dimension were considered. In general, the (possibly infinite) uniform dimension of a module M is defined to be the supremum k of all cardinalities of independent families of submodules and one says that the uniform dimension is attained if M contains an independent family of non-zero submodules of cardinality k. John Dauns and Laszlo Fuchs proved that any uniform dimension that is not an inaccessible cardinality is attained. Together with Edmund Puczylowski, we show that the gap in Camillo and Zelmanowitz proof can be fixed in case the uniform dimension k of R is attained and R is either commutative or R has a base, i.e. it contains an independent family of k uniform(!) submodules.

Date:  2019-05-29
Start Time:   15:00
Speaker:  Christian Lomp (Univ. Porto)
Institution:  Universidade do Porto
Place:  Sala 5.5
Research Groups: -Algebra and Combinatorics
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