Enumeration of subrings and subring identities of strongly unital commutative finite rings

Daisy Ingado Binayo; Michael Onyango Ojiema; Maurice Owino Oduor

doi:10.51867/asarev.3.1.15

Authors

Daisy Ingado Binayo Department of Mathematics, Masinde Muliro University of Science and Technology, Kenya Author https://orcid.org/0009-0009-1132-0038
Michael Onyango Ojiema Department of Mathematics, Masinde Muliro University of Science and Technology, Kenya Author https://orcid.org/0000-0001-9635-7597
Maurice Owino Oduor Department of Mathematics and Computer Science, University of Kabianga, Kenya Author

DOI:

https://doi.org/10.51867/asarev.3.1.15

Keywords:

Finite Commutative Rings, Ring Enumeration, Square-Free Integers, Strongly Unital Finite Rings

Abstract

In this paper, we derive explicit formulas for the number of subrings of a given strongly unital ring and for the identities of those subrings. From the classification results, every finite commutative strongly unital ring R ≅ ∏ᵢ₌₁ᵏ ℤₚᵢ, where k ≥ 2 and p₁, …, pₖ are distinct primes. Equivalently, by the Chinese Remainder Theorem, R ≅ ℤₙ, where n = p₁p₂···pₖ is square-free and composite. This paper provides closed-form formulas for the number of subrings, describes the subrings explicitly, identifies the identity element of each subring, and discusses their combinatorial and algorithmic aspects.

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