The dual-classical Krull dimension of rings via topology

Let R be a ring and \mathcal{X} = \mathcal{SH}(R)-\{۰\} be the set all of non-zero strongly hollow ideals (briefly, sh-ideals) of R. We first study the concept SH-topology and investigate some of the basic properties of a topological space with this topology. It is shown that, if \mathcal X is with SH-topology, then \mathcal {X} is Noetherian if and only if every subset of \mathcal X is quasi-compact if and only if R has dcc on semi-sh-ideals. Finally, the relation between the dual-classical Krull dimension of R and the derived dimension of \mathcal {X} with a certain topology has been studied. It is proved that, if \mathcal {X} has derived dimension, then R has the dual-classical Krull dimension and in case R is a D-ring (i.e., the lattice of ideals of R is distributive), then the converse is true. Moreover these two dimension differ by at most ۱.

کلیدواژه ها:

Strongly hollow ideals ، SH-topology ، derived dimension ، dual-classical Krull dimension