TY - JOUR
AU - Lenagan, T. H.
AB - T. H. LENAGAN In order to compute the Krull dimension of a module it is useful to have a comparison between the Krull dimensions of the module and its submodules. One of the basic results in [1] is that a module with Krull dimension which is the sum of submodules of Krull dimension ^ a has itself got Krull dimension ^ a. If N is a submodule of a module M with Krull dimension then kd(M) = sup (kd(M/JV), kd(N)) and from this the result follows easily for finite sums of submodules. However, for infinite sums the solution is not so simple and one is forced to consider the Krull dimension of modules which arise from infinite ascending chains of submodules of Krull dimension < a. Gordon and Robson's proof of this result is complicated and divides into two distinct cases. First, if the Krull dimension, a, is not a limit ordinal, then the proof is essentially the same as in the Krull dimension zero (or Artinian) case where it is only necessary to prove that a module is Artinian if it is a sum of Artinian submodules and every factor module has finite uniform dimension. The second case, 
TI - Modules with Krull Dimension
JO - Bulletin of the London Mathematical Society
DO - 10.1112/blms/12.1.39
DA - 1980-01-01
UR - https://www.deepdyve.com/lp/wiley/modules-with-krull-dimension-mMWBO3VloH
SP - 39
EP - 40
VL - 12
IS - 1
DP - DeepDyve
ER -