2011/10/11 by Li, Huishi
#16W50 #16W70 #16Z05 #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1110.2248
Let K =K be the free K-algebra on X=X1,...,Xn over a field K, which is equipped with a weight ℕ-gradation (i.e., each Xi is assigned a positive degree), and let \cal G be a finite homogeneous Gröbner basis for the ideal I= of K with respect to some monomial ordering \prec on K. It is proved that if the monomial algebra K/ is semi-prime, where \bf LM(\cal G) is the set of leading monomials of \cal G with respect to \prec, then the ℕ-graded algebra A=K/I is semiprimitive (in the sense of Jacobson). In the case that \cal G is a finite non-homogeneous Gröbner basis with respect to a graded monomial ordering \precgr, and the ℕ-filtration FA of the algebra A=K/I induced by the ℕ-grading filtration FK of K is considered, if the monomial algebra K/ is semi-prime, then it is proved that the associated ℕ-graded algebra G(A) and the Rees algebra \widetildeA of A determined by FA are all semiprimitive.