Article Sidebar
Published:
Jun 28, 2026Keywords:
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
(i) AJER is an open-access journal. This means that all content is freely available to all. The author retains the copyrights associated with their work within AJER’s policy framework.
(ii) All articles can be accessed freely without any limitations of use, distribution, and reproduction in any medium so long as they are expressly cited.
(iii) The use of general descriptive names, trade names, trademarks, and so forth in this publication, even if not specifically identified, may not mean that these names are not protected by the relevant laws and regulations in their respective countries.
(iv) The publisher believes that all information is true and accurate at the time of going to press. In this regard, the authors, editors, and publishers cannot accept any legal responsibility for any errors, typos or omissions that may be made. There is thus no warranty for the materials contained in the articles.
Main Article Content
Construction and classification of strongly unital commutative finite rings
Daisy Ingado Binayo
Michael Onyango Ojiema
Maurice Owino Oduor
Abstract
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.
