Construction and classification of strongly unital commutative finite rings

Daisy Ingado Binayo; Michael Onyango Ojiema; Maurice Owino Oduor

doi:10.51867/ajernet.maths.7.2.132

Autores

Daisy Ingado Binayo Department of Mathematics, Masinde Muliro University of Science and Technology, Kenya https://orcid.org/0009-0009-1132-0038
Michael Onyango Ojiema Department of Mathematics, Masinde Muliro University of Science and Technology, Kenya https://orcid.org/0000-0001-9635-7597
Maurice Owino Oduor Department of Mathematics and Computer Science, University of Kabianga, Kenya https://orcid.org/0009-0005-7725-5487

DOI:

https://doi.org/10.51867/ajernet.maths.7.2.132

Palavras-chave:

Classification of Commutative Rings, External and Internal Structures of Finite Rings, Strongly Unital Finite Rings

Resumo

This paper investigates finite commutative strongly unital rings, a class of rings in which every proper nontrivial subring possesses a multiplicative identity distinct from that of the ambient ring and from the identities of all other subrings. The study is motivated by the observation that, although finite commutative unital rings have been classified as direct products of fields of prime order, proper subrings may share the same identity as the ambient ring. To address this limitation, the notion of strong unitality is introduced and developed. General classes of strongly unital rings are constructed using direct products of fields of distinct prime characteristics. Necessary and sufficient conditions for strong unitality are established through the behavior of subring identities and idempotent elements. It is shown that finite commutative strongly unital rings admit a highly restrictive structure determined by distinct prime field components. Furthermore, a complete classification of finite commutative strongly unital rings is obtained. In particular, it is proved that a finite commutative ring is strongly unital if and only if it is isomorphic to a finite direct product of fields of distinct prime orders. Consequently, every finite commutative strongly unital ring is characterized up to isomorphism by the set of distinct primes appearing in its decomposition. The results provide both a constructive framework and a complete structural characterization of finite commutative strongly unital rings.

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