Specht property for weak graded identities of matrix algebras of order two

Abstract Let D be an infinite commutative unital domain with $$\textrm{char}\,D\ne 2$$ char D ≠ 2 . We study graded polynomial identities of the pair $$(M_2(D),sl_2(D))$$ ( M 2 ( D ) , s l 2 ( D ) ) , where $$sl_2(D)$$ s l 2 ( D ) is considered either as a graded D -submodule or a Lie subalgebra of $$M_2(D)^{(-)}$$ M 2 ( D ) ( - ) . According to the structure considered, we have different notions of the consequences of identities: L -consequences, when $$sl_2(D)$$ s l 2 ( D ) is viewed as a Lie algebra, and S -consequences, when it is viewed as a submodule. Our main result is that if D is Noetherian, then for any non-trivial G -grading the S –weak ideal of graded identities of $$(M_2(D),sl_2(D))$$ ( M 2 ( D ) , s l 2 ( D ) ) satisfies the Specht property. We also prove that the notion of S -consequences is “stronger” than the notion of L -consequences. There are, up to equivalence, three gradings by the groups $$\mathbb {Z}_2$$ Z 2 , $$\mathbb {Z}$$ Z and $$\mathbb {Z}_2\times \mathbb {Z}_2$$ Z 2 × Z 2 . For each of these gradings, we provide a finite basis of the weak graded identities of