Algebras and their Associated Monomial Algebras

Abstract

Let $R=\oplus_{Γ\inΓ}R_γ$ be a $Γ$-graded $K$-algebra over a field $K$, where $Γ$ is a totally ordered semigroup, and let $I$ be an ideal of $R$. Considering the $Γ$-grading filtration $FR$ of $R$ and the $Γ$-filtration $FA$ induced by $FR$ for the quotient $K$-algebra $A=R/I$, we show that there is a $Γ$-graded $K$-algebra isomorphism $G(A)\cong \bar{A}=R/<\hbox{\bf HT} (I)>$, where $G(A)$ is the associated $Γ$-graded $K$-algebra of $A$ defined by $FA$, and $<\hbox{\bf HT}(I)>$ is the $Γ$-graded ideal of $R$ generated by the set of head terms of $I$. In the case that $Γ$ is an ordered monoid with a well-ordering, this result enables us to lift many nice structural properties of $\bar{A}$ to $A$ theoretically, and the natural connection with Gröbner basis theory leads to effective realization lifting information from the associated monomial algebras in both commutative and noncommutative cases.