Let F be a field of characteristic zero and let R be an algebra that admits a regular grading by an abelian group H. Moreover, we consider G a group and let A be an algebra with a grading by the group G × H, we define the R-hull of A as the G × H-graded algebra given by \(\mathfrak {R}(A)=\oplus _{(g,h)\in G\times H}A_{(g,h)}\otimes R_{h}\). In this paper we provide a basis for the graded identities (resp. central polynomials) of the R-hull of A, assuming that a (suitable) basis for the graded identities (resp. central polynomials) of the G × H-graded algebra A is known. In particular, for any a, \(b\in \mathbb {N}\), we find a basis for the graded identities and the graded central polynomials for the algebra M_a,b(E), graded by the group \(G\times \mathbb {Z}_{2}\). Here E is the Grassmann algebra of an infinite dimensional F-vector space, equipped with its natural \(\mathbb {Z}_{2}\)-grading and the matrix algebra M_a+b(F) is equipped with an elementary grading by the group \(G\times \mathbb {Z}_{2}\), so that its neutral component coincides with the subspace of the diagonal matrices. We describe the isomorphism classes of gradings on M_a,b(E) that arise in this way and count the isomorphism classes of such gradings. Moreover, we give an alternative proof of the fact that the tensor product M_a,b(E) ⊗ M_r,s(E) is PI-equivalent to M_ar+bs,as+br(E). Finally, when the grading group is \(\mathbb {Z}_{3}\times \mathbb {Z}_{2}\) (resp. \(\mathbb {Z}\times \mathbb {Z}_{2}\)), we present a complete description of a basis for the graded central polynomials for the algebra M_2,1(E) (resp. M_a,b(E) in the case b = 1).

令F是一个特征为零的域，让R是一个代数，它允许一个阿贝尔群H的规则分级。此外，我们考虑ģ的基团，并让阿是与由组分级的代数ģ × ħ，我们定义了ř的-hull甲作为ģ × ħ -graded代数由下式给出\（\ mathfrak {R}（A )=\oplus _{(g,h)\in G\times H}A_{(g,h)}\otimes R_{h}\)。在本文中，我们为A的R壳的分级恒等式（或中心多项式）提供了基础，假设G × H分级代数 A的分级恒等式（中心多项式）的（合适的）基是已知的。特别地，对于任何a , \(b\in \mathbb {N}\)，我们找到了代数M _{a , b} ( E )的分级恒等式和分级中心多项式的基，由组\( G\times \mathbb {Z}_{2}\)。这里E是无限维F向量空间的格拉斯曼代数，配备其自然\(\mathbb {Z}_{2}\) -分级和矩阵代数M _{a + b}( F ) 由群\(G\times \mathbb {Z}_{2}\)配备了初等分级，因此其中性分量与对角矩阵的子空间重合。我们描述以这种方式出现的M _{a , b} ( E ) 上分级的同构类，并计算这些分级的同构类。此外，我们给出了一个替代证明，即张量积M _{a , b} ( E ) ⊗ M _{r , s} ( E ) 是 PI 等价于M _{a r + b s，a s + b r} ( E )。最后，当评分组为\(\mathbb {Z}_{3}\times \mathbb {Z}_{2}\) (resp. \(\mathbb {Z}\times \mathbb {Z}_{ 2}\) )，我们提供了代数M _2,1 ( E )的分级中心多项式的基础的完整描述（在b = 1的情况下_，分别为M _{a , b} ( E )）。