Classify finite commutative local rings of order p^n

Classify all commutative local rings of order p^n for a given prime p; that is, determine a complete classification of finite commutative local rings whose cardinality equals p^n.

Background

The paper focuses on computing the Schur multiplier of the special linear group SL_2 over finite commutative rings and reduces the paper to finite commutative local rings. While it is straightforward that such rings have order a power of a prime, a comprehensive classification of finite commutative local rings of a fixed order p^n is acknowledged to be a difficult problem.

This classification problem is highlighted as a gap in current knowledge and is relevant to the structural understanding of finite local rings, which in turn supports computations of group homology and algebraic K-theory central to the paper's results.