Universal enveloping algebra for abelian Lie algebra equals symmetric algebra on underlying module

Statement for fields
Suppose $$K$$ is a field and $$L$$ is an abelian Lie algebra over $$K$$. The universal enveloping algebra $$U(L)$$ is a commutative associative unital $$K$$-algebra defined as the symmetric algebra over $$L$$, viewed as a $$K$$-vector space.

Statement for commutative unital rings
Suppose $$R$$ is a commutative unital ring and $$L$$ is an abelian Lie algebra over $$R$$. The universal enveloping algebra $$U(L)$$ is a commutative associative unital $$R$$-algebra defined as the symmetric algebra over $$L$$, viewed as a $$R$$-module.

Statement for Lie rings as algebras over the integers
Suppose $$L$$ is an abelian Lie ring, viewed as a $$\mathbb{Z}$$-Lie algebra. The universal enveloping algebra $$U(L)$$ is a commutative associative unital ring defined as the symmetric algebra of $$L$$ viewed as an abelian group.