  Let $D$ be a Noetherian infinite integral domain, denote by $M_2(D)$ and by
$sl_2(D)$ the $2\times 2$ matrix algebra and the Lie algebra of the traceless
matrices in $M_2(D)$, respectively. In this paper we study the natural grading
by the cyclic group $\mathbb{Z}_2$ of order 2 on $M_2(D)$ and on $sl_2(D)$. We
describe a finite basis of the graded polynomial identities for the pair
$(M_2(D), sl_2(D))$. Moreover we prove that the ideal of the graded identities
for this pair satisfies the Specht property, that is every ideal of graded
identities of pairs (associative algebra, Lie algebra), satisfying the graded
identities for $(M_2(D), sl_2(D))$, is finitely generated. The polynomial
identities for $M_2(D)$ are known if $D$ is any field of characteristic
different from 2. The identities for the Lie algebra $sl_2(D)$ are known when
$D$ is an infinite field. The identities for the pair we consider were first
described by Razmyslov when $D$ is a field of characteristic 0, and afterwards
by the second author when $D$ is an infinite field. The graded identities for
the pair $(M_2(D), gl_2(D))$ were also described, by Krasilnikov and the second
author.
  In order to obtain these results we use certain graded analogues of the
generic matrices, and also techniques developed by G. Higman concerning
partially well ordered sets.
