Primeness property for regular gradings

Abstract

Let $K$ be an algebraically closed field of characteristic $0$ and $G$ a finite abelian group. For a $G$-graded $K$-algebra $A$, we define the primeness property for graded central polynomials: for any graded polynomials $f$ and $g$ in disjoint sets of variables, if $fg$ is graded central, then both $f$ and $g$ are graded central. Let $A=\bigoplus_{g\in G} A_g$ be its decomposition into homogeneous components. Assume that for every $n$-tuple $(g_1,\dots,g_n)$ in $G$, there exist $a_{i}\in A_{g_{i}}$ with $a_1\cdots a_n\neq 0$, and that for each $g$,$h\in G$ there exists a scalar $\beta(g,h)\in K^{\ast}$ such that $a_ga_h=\beta(g,h)a_ha_g$. Then the grading is regular, and minimal if no distinct $g$, $h\in G$ satisfy $\beta(g,x)=\beta(h,x)$ for all $x\in G$. We prove that $G$-graded regular algebras, including $M_n(K)$ with the Pauli grading, fail the primeness property. For matrices of orders $2$ and $3$, no nontrivial gradings satisfy primeness. Finally, for $\mathbb{Z}_2$-graded regular algebras, we use the known fact that minimal regular gradings satisfy the graded identities of the infinite-dimensional Grassmann algebra $E$ and contain a copy of $E$ to show that such algebras satisfy the primeness property in the ordinary sense. As a consequence, we show that minimality is not required for the regularity of the grading.