I'm now interested in classifying the indecomposable modules over $\mathbb{Z}/p^{2}\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}]$ : the group ring of $\mathbb{Z}/p\mathbb{Z}$ over the ring $\mathbb{Z}/p^{2}\mathbb{Z}$.


I think that a piece of information such as an upper bound for the number of indecomposable modules whose $p$-rank is bounded by $m$ will be very helpful for my research.


Before I try by myself, I would like to hear about any known methods to classify indecomposable modules over algebra that are not necessarily over a field. Since the ring is not complicated, I hope there can be some answer.


Any references are welcome. Since I'm not an expert in the pure ring theory, I'm not familiar with many concepts like a serial ring, perfect ring,... etc. If you could suggest some ring-theoretic concepts satisfied by the ring above and which might be helpful, then it would be a great help to me.


Thank you very much in advance.



Classifying indecomposable modules over $\mathbb{Z}/p^{2}\mathbb{Z}[\mathbb{Z}/p\mathbb{Z}]$
This was done by Szekeres, "Determination of a certain family of finite metabelian groups" Trans. Amer. Math. Soc. 66 (1949), 1–43.


See also Nazarova & Roiter, "Finitely generated modules over a dyad of two local Dedekind rings, and finite groups which possess an abelian normal divisor of index $p$" Math USSR Izvestija 3 (1969), no. 1, 65–86, and


Aviño-Diaz & Bautista Ramon, "The additive structure of indecomposable $Z\_{p^n}C\_p$-modules" Comm. Algebra 24 (1996), no. 8, 2567–2595.


The answers are not easy to state, let alone prove, so good luck.


