@article{Alabiad2022, 
author = {Sami Alabiad and Yousef Alkhamees},
title = {On classification of finite commutative chain rings},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {2},
pages = {1742-1757},
keywords = {finite chain rings, p-adic fields, Galois rings, j-diagram, isomorphism class},
url = {https://www.sciopen.com/article/10.3934/math.2022100},
doi = {10.3934/math.2022100},
abstract = {Let    R be a finite commutative chain ring with invariants    p  ,  n  ,  r  ,  k  ,  m  . It is known that    R is an extension over a Galois ring    G  R  (      p    n    ,  r  ) by an Eisenstein polynomial of some degree    k. If    p  ∤  k  , the enumeration of such rings is known. However, when    p  ∣  k, relatively little is known about the classification of these rings. The main purpose of this article is to investigate the classification of all finite commutative chain rings with given invariants    p  ,  n  ,  r  ,  k  ,  m up to isomorphism when    p  ∣  k  . Based on the notion of j-diagram initiated by Ayoub, the number of isomorphism classes of finite (complete) chain rings with    (  p  −  1  )  ∤  k is determined. In addition, we study the case    (  p  −  1  )  ∣  k  , and show that the classification is strongly dependent on Eisenstein polynomials not only on    p  ,  n  ,  r  ,  k  ,  m  . In this case, we classify finite (incomplete) chain rings under some conditions concerning the Eisenstein polynomials. These results yield immediate corollaries for p-adic fields, coding theory and geometry.}
}