Graded polynomial identities and central polynomials of matrices over an infinite integral domain

Let $K$ be an infinite integral domain and $M_n(K)$ be the algebra of all $n\times n$ matrices over $K$. This paper aims for the following goals: Find a basis for the graded identities for elementary grading in $M_n(K)$ when the neutral component and diagonal coincide; Describe the ${\mathbb{Z}}_p$-graded central polynomials of $M_p(K)$ when $p$ is a prime number; Describe the $\mathbb{Z}$-graded central polynomials of $M_n(K)$.