Length of the group algebra of the direct product of a cyclic group and a cyclic $p$-group in the modular case

In this paper, the length of the group algebra of the direct product of a cyclic group and a cyclic $p$-group over a field of characteristic $p$ is calculated. A general lower bound for the length of a commutative group algebra is proved, and in the case of the direct product of a cyclic group and a cyclic $p$-group this bound is sharp.