Some Results on dh-Closed Homogeneous Gröbner Bases and dh-Closed Graded Ideals

Let K be a field and R = ⊕p∈NRp an N-graded K-algebra, which has an SM K-basis (i.e. a skew multiplicative K-basis) such that R holds a Gröbner basis theory. It is proved that there is a one-to-one correspondence between the set of Gröbner bases in R and the set of dh-closed homogeneous Gröbner bases in the polynomial algebra R[t]; and that the similar result holds true if R and R[t] are replaced respectively by the free algebra K〈X1, ..., Xn〉 and the free algebra K〈X1, ..., Xn, T 〉. Moreover, it is shown that dh-closed graded ideals in R[t] and K〈X1, ..., Xn, T 〉 can be realized by dh-closed homogeneous Gröbner bases. The latter result indeed tells us that algebras defined by dh-homogeneous Gröbner bases can be studied as Rees algebras effectively via more simpler algebras as demonstrated in ([7], [8]). 2000 Mathematics Classification Primary 16W70; Secondary 68W30 (16Z05).