2009/07/03 by Li, Huishi, Su, Cang
#16W70 #16Z05 #68W30 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.0907.0526
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]).