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For any differential graded (DG for short) Poisson algebra $A$ given bygenerators and relations, we give a "formula" for computing the universalenveloping algebra $A^e$ of $A$. Moreover, we prove that $A^e$ has aPoincar\'e-Birkhoff-Witt basis provided that $A$ is a graded commutativepolynomial algebra. As an application of the PBW-basis, we show that a DGsymplectic ideal of a DG Poisson algebra $A$ is the annihilator of a simple DGPoisson $A$-module, where $A$ is the DG Poisson homomorphic image of a DGPoisson algebra $R$ whose underlying algebra structure is a graded commutativepolynomial algebra.

PBW-basis for universal enveloping algebras of differential graded Poisson algebras