In the first chapter some basic notions and results from commutative algebra are being introduced. In the second chapter we present new results related to sequentially Cohen Macaulay, pretty clean and Stanley ideals. Let I be a monomial ideal of the poly- nomial ring S. We show that S=I is sequentially Cohen-Macaulay if and only if S=I is pretty clean. In particular, if S=I is sequentially Cohen- Macaulay then I is a Stanley ideal. Chapter 3 includes some results related to Stanley Conjecture. We define nice partition of the multicomplex associated to a Stanley ideal. In the main result we show that if the multicomplex associated to monomial ideal I has a nice partition then the multicomplex associated to polarized ideal Ip has a nice partition. Fourth and last chapter deals with the regularity of ideals of Borel type. We have proved that the regularity of monomial ideals whose associated prime ideals are totally ordered by inclusion is linearly bounded.
In the first chapter some basic notions and results from commutative algebra are being introduced along with a description on the progress towards the Stanley s conjecture. In the second chapter, we have discussed the Stanley s conjecture for a polynomial ring over a field in four variables. In the case of polynomial ring in five variables, we have presented that the monomial ideals with all associated primes of height two, are Stanley ideals. Moreover, we have introduced the Janet s algorithm for the Stanley decomposition of a monomial ideal and discussed the Janet s algorithm for the squarefree Stanley decomposition of squarefree monomial ideal. We conclude the chapter with the discussion of the Janet s algorithm for the partition of a simplicial complex. In the third and last chapter, we have discussed the regularity of monomial ideals in a polynomial ring in n variables, whose associated prime ideals are totally ordered by inclusion.