Recognizing The Semiprimitivity of $\mathbb{N}$-graded Algebras via Gr\"obner Bases

Let $K<X> =K<X_1,...,X_n>$ be the free $K$-algebra on $X={X_1,...,X_n}$ over a field $K$, which is equipped with a weight $\mathbb{N}$-gradation (i.e., each $X_i$ is assigned a positive degree), and let ${\cal G}$ be a finite homogeneous Gr\"obner basis for the ideal $I=<{\cal G}>$ of $K<X>$ with respect to some monomial ordering $\prec$ on $K<X>$. It is proved that if the monomial algebra $K<X>/<{\bf LM}({\cal G})>$ is semi-prime, where ${\bf LM}({\cal G})$ is the set of leading monomials of ${\cal G}$ with respect to $\prec$, then the $\mathbb{N}$-graded algebra $A=K<X>/I$ is semiprimitive (in the sense of Jacobson). In the case that ${\cal G}$ is a finite non-homogeneous Gr\"obner basis with respect to a graded monomial ordering $\prec_{gr}$, and the $\mathbb{N}$-filtration $FA$ of the algebra $A=K<X>/I$ induced by the $\mathbb{N}$-grading filtration $FK<X>$ of $K<X>$ is considered, if the monomial algebra $K<X>/<{\bf LM}({\cal G})>$ is semi-prime, then it is proved that the associated $\mathbb{N}$-graded algebra $G(A)$ and the Rees algebra $\widetilde{A}$ of $A$ determined by $FA$ are all semiprimitive.