|
Abstract:
|
This dissertation examines simplicial complexes associated with commutative
rings with unity. In general, a combinatorial object can be attached to a ring in
many di erent ways, and in this dissertation we examine several simplicial complexes
attached to rings which give interesting results. Focus of this thesis is determining
the homotopy type of geometric realization of these simplicial complexes, when it is
possible.
For a partially ordered set of nontrivial ideals in a commutative ring with identity,
we investigate order complex and determine its homotopy type for the general case.
Simplicial complex can also be associated to a ring indirectly, as an independence
complex of some graph or hypergraph which is associated to that ring. For
the comaximal graph of commutative ring with identity we de ne its independence
complex and determine its homotopy type for general commutative rings with identity.
This thesis also focuses on the study of zero-divisors, by investigating ideals
which are zero-divisors and de ning zero-divisor ideal complex. The homotopy type
of geometric realization of this simplicial complex is determined for rings that are
nite and for rings that have in nitely many maximal ideals. In this part of the
thesis, we use the discrete Morse theory for simplicial complexes. The theorems
proven in this dissertation are then applied to certain classes of commutative rings,
which gives us some interesting combinatorial results. |