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Let $H$ be a finite dimensional pointed rank one Hopf algebra of nilpotenttype. We first determine all finite dimensional indecomposable $H$-modules upto isomorphism, and then establish the Clebsch-Gordan formulas for thedecompositions of the tensor products of indecomposable $H$-modules by virtueof almost split sequences. The Green ring $r(H)$ of $H$ will be presented interms of generators and relations. It turns out that the Green ring $r(H)$ iscommutative and is generated by one variable over the Grothendieck ring$G_0(H)$ of $H$ modulo one relation. Moreover, $r(H)$ is Frobenius andsymmetric with dual bases associated to almost split sequences, and itsJacobson radical is a principal ideal. Finally, we show that the stable Greenring, the Green ring of the stable module category, is isomorphic to thequotient ring of $r(H)$ modulo the principal ideal generated by the projectivecover of the trivial module. It turns out that the complexified stable Greenalgebra is a group-like algebra and hence a bi-Frobenius algebra.

Green Rings of Finite Dimensional Pointed Rank One Hopf algebras of Nilpotent Type