Characterisation of b-semiring by Weak Ideals under ⋆ Product and o Product
Muddsar Yaseen Malik
*
Department of Mathematics, Yenepoya Institute of Technology, Thodar, Modbidri, 574225, Mangaluru, India.
D. Sivakumar
Department of Mathematics, Annamalai University, Annamalainagar, Tamil Nadu, India.
*Author to whom correspondence should be addressed.
Abstract
In this paper, weak-1 right ideals (W1RI), weak-2 right ideals (W2RI), weak-1 left ideals (W1LI), and weak-2 left ideals (W2LI) generated by a single element are introduced and studied in the setting of b-semirings. The basic properties of these one-sided weak ideals are examined, together with the relationships among the different classes of weak ideals. Particular attention is given to the two product operations defined on b-semirings and to the behaviour of weak ideals under these operations. The generated forms of weak-1 right and weak-1 left ideals are described, and auxiliary results concerning products of subsets are established. The products of weak ideals with right and left ideals are then investigated systematically. Conditions are obtained under which such products form weak-1 ideals, weak-2 ideals, right ideals, or left ideals, as appropriate. The resulting inclusion and closure relations are used to connect the algebraic structure of a b-semiring with the behaviour of its one-sided weak ideals and their products. Several examples and counter examples are included to distinguish the different cases and to show that certain converse statements do not hold. Overall, the study provides product-based structural characterisations of b-semirings using weak-1 and weak-2 one-sided ideals without extending the analysis beyond the ideal classes and operations considered in the manuscript.
Keywords: b-semiring, weak-1 right ideal, weak-1 left ideal, weak-2 right ideal, weak-2 left ideal, one-sided weak ideals, ideal products, generated weak ideals, structural characterisation, algebraic ideals