Good evening,
$R$ is a commutative ring and $A$ is a commutative $R$-algebra. Let $(L,\rho)$ a $(R,A)$-Lie Rinehart algebra. $L$ is a $R$-Lie algebra with a $A$-module structure and $\rho:L\longrightarrow Der_R(A)$ is a morphism of $R$-Lie algebra and of $A$-modules. We have the property $$[x,ay]=\rho(x)(a)y+a[x,y]$$
I consider $U(L)$ the universal envelopping algebra of $L$. Generaly, $U(L)\otimes_A U(L)$ is not an algebra with the usual rule for tensor product of algebras. i.e: the product $$(u\otimes v)(u'\otimes v')=uu'\otimes vv'$$ is not defined on $U(L)\otimes_AU(L)$. I don't understand why. I would be gratefull if sommeone can help or give me a reference.
Thank you.