Ring R (?) M The K-gorenstein And Cohomology

In this thesis, we will discuss the homological dimension of the ring R × M , and we also investigate its A:-Gorenstein property . Let R be an associative ring with identity, M a R-R-bimodule,R × M = {(r × m)|r ∈ R, m ∈ M}. For any (r1 × m1), (r2 × m2) ∈ R × M, we define(r1 × m1) + (r2 × m2) = ((r1 + r2) × (m1 + m2)); (r1 × m1)(r2 × m2) = ((r1r2) × (r1m2 + m1r2)).Since the K-Gorenstein property of ring R x M is an important aspect in the research field, in the first chapter, we have got an equivalent condition for R × M as a K-Gorenstein ring by study the injective resolution of ring R × M.The dimensions of rings is one of the most important parts in homological theory. Given a ring, one always takes the supremum of some homological dimension of specified modules to obtain the corresponding global dimension, and to characterize the ring from outside. Given a ring or module , one can define various homological dimensions by resolving the modules. In the second chapter, we attain this goal by another route. Collecting all short exact sequence and the morphisms among them, we get a new category, call The Short Exact Sequences Category CRM. We define a global dimension attached to the original ring R from the view of The Short Exact Sequences Category CR.M, named The Exact Projective Dimension. At the end of this chapter, we get some results about the dimension and its relationship with global dimension of a ring R.In the third chapter, the homological dimensions about ring R × M have be discussed. [5] have enlighten us that there is a solid connection between R×M and so called Morita System Ring. But few papers published about the global dimension of Morita System on the universal condition, because it is so complex that it have only be discussed under a few strict conditions. The Literature [5] and some other ones hold that Morita system play as the trivial extension of R × M , however, no more future result have be seen since 70s' by the route of trivial extension. In this chapter, we estimnated the global dimension of R × M byan exact sequence which have four items. By using the well-known dimension shifting equation, we get:Let R be a left-noether ring, M be a Noetherian, M be flat right module. Then we haveWe give the exact protective dimension of R ×M in this chapter as well.