Definition:Linear Form

Let $$R$$ be a commutative ring.

Let $$\left({G, +_G: \circ}\right)_R$$ be a module over $$R$$.

Let $$\phi: \left({G, +_G: \circ}\right)_R \to \left({R, +_R: \circ}\right)_R$$ be a linear transformation from $$G$$ to the $R$-module $R$.

Then $$\phi$$ is called a linear form on $$G$$ (or linear functional).