Indeterminacy of Boolean Ring

<ns3:p>Background A neutrosophic ring represents an algebraic generalization of the classical ring structure by introducing an indeterminacy element I , enabling the modeling of truth, falsity, and indeterminacy simultaneously, as established within Smarandache’s neutrosophic framework. In contrast, a Boolean ring is a commutative algebraic structure in which every element is idempotent ( αI ) ² = αI reflecting the logical principles of Boolean algebras and possessing characteristic two Combining these concepts, the neutrosophic Boolean ring extends the Boolean ring by embedding neutrosophic logic parameters—truth (T), indeterminacy (I), and falsity (F)—into its elements and operations. This hybrid structure allows for the representation of algebraic uncertainty and incomplete information while preserving Boolean idempotent properties, thus providing a flexible framework for studying systems with uncertain or partially defined information in algebraic and logical contexts Methods The research defines the Indeterminacy ring R I = { α + βI : α , β ∈ R } and explores its algebraic properties through examples from integers, rationals, and reals. It then formulates the Indeterminacy Boolean Ring (B-Ring) characterized by idempotency ( αI ) ² = αI , and establishes several theorems proving its core algebraic features. Results Findings reveal that Indeterminacy B-Rings are commutative and have characteristic two, ensuring 2 αI = 0 . Each maximal Indeterminacy ideal is also prime, and these rings are semisimple and reduced, containing no nonzero nilpotent elements. Furthermore, any Indeterminacy B-Ring can be represented as a direct product of copies of Z ₂ I , known as the Indeterminacy Boolean field. The quotient rings preserve Boolean and Indeterminacy properties, confirming their structural consistency. Conclusions The study successfully extends Boolean ring theory to the Indeterminacy domain, establishing a strong algebraic foundation for modeling uncertainty. Indeterminacy B-Rings maintain the essential Boolean properties of idempotency and commutativity while incorporating indeterminate behavior through I ² = I . These results open new perspectives for future applications in neutrosophic logic, fuzzy systems, and abstract algebra dealing with indeterminate information.</ns3:p>