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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 primenumber; Describe the $\mathbb{Z}$-graded central polynomials of $M_{n}(K)$.

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