Primeness property for regular gradings

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 $β(g,h)\in K^{\ast}$ such that $a_ga_h=β(g,h)a_ha_g$. Then the grading is regular, and minimal if no distinct $g$, $h\in G$ satisfy $β(g,x)=β(h,x)$ 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.

💡 Research Summary

The paper investigates the “primeness property” for graded central polynomials in the setting of finite‑abelian‑group gradings on associative algebras over an algebraically closed field (K) of characteristic zero. A graded central polynomial for a (G)-graded algebra (A=\bigoplus_{g\in G}A_g) is a polynomial that, after any homogeneous substitution, yields an element of the centre (Z(A)). The primeness property asks that if two graded polynomials (f) and (g) involve disjoint sets of variables and their product (fg) is a (proper) graded central polynomial, then each of (f) and (g) must already be a graded central polynomial.

The authors focus on regular gradings, defined by two conditions: (i) for any finite tuple ((g_1,\dots,g_n)) there exist homogeneous elements (a_i\in A_{g_i}) with non‑zero product, and (ii) there is a bicharacter (\beta:G\times G\to K^\ast) such that (a_g a_h=\beta(g,h)a_h a_g) for all homogeneous (a_g\in A_g), (a_h\in A_h). A regular grading is minimal when distinct group elements give distinct rows of the matrix ((\beta(g,h))).

The main result is that any algebra equipped with a regular grading fails the primeness property for graded central polynomials. The proof proceeds by first showing that any regular grading can be coarsened to a minimal regular grading via the subgroup (G_0={g\mid\beta(g,h)=1\ \forall h}). In the minimal setting one can explicitly construct two non‑central graded polynomials whose product becomes central, thereby violating primeness. Consequently, the property cannot hold for any regular grading.

Applying this to matrix algebras, the authors consider the Pauli grading on (M_n(K)) (the grading induced by the bicharacter of (\mathbb Z_2\times\mathbb Z_2)). Since this grading is regular, (M_n(K)) does not satisfy the primeness property for graded central polynomials. Moreover, a detailed analysis for (n=2) and (n=3) shows that no non‑trivial grading on these algebras enjoys the primeness property; the authors conjecture the same for all (n\ge4).

The paper then turns to (\mathbb Z_2)-graded regular algebras and the ordinary (non‑graded) primeness property. It is known that a minimal (\mathbb Z_2)-regular grading forces the algebra to satisfy the same graded identities as the infinite‑dimensional Grassmann algebra (E) and to contain a copy of (E). Since (E) possesses the primeness property for its (ordinary) central polynomials, any (\mathbb Z_2)-regular algebra inherits this property. Importantly, the authors demonstrate that the minimality assumption is unnecessary: regularity alone suffices for the ordinary primeness property in the (\mathbb Z_2) case.

In summary, the work establishes a clear dichotomy: regular gradings destroy the primeness property for graded central polynomials, yet in the special case of (\mathbb Z_2) regular gradings the ordinary primeness property survives, independent of minimality. These findings deepen the understanding of how group gradings interact with polynomial identities and central polynomials, and they provide concrete counterexamples (e.g., Pauli‑graded matrix algebras) as well as positive results for algebras closely related to the Grassmann algebra. The paper contributes both to the structure theory of graded PI‑algebras and to the broader program of classifying verbally prime algebras under various grading schemes.