Rings with Commutative Multiplication on One-sided Ideals and Algebraic Structure for Lattices of Submodules

Numerous studies on associative rings exhibiting certain commutative behaviours have been conducted over the past half-century. One of the motivations in this area is to identify noncommutative rings that retain some useful properties of commutative rings. In this thesis, we introduce and study a new class of rings that retains abundant properties of commutative rings. Although the rings are not necessarily commutative, they exhibit striking similarities to commutative rings and form a class of rings which is “closer” to the class of commutative rings in comparison to the well-known class of left duo rings. We generalise several classical results from commutative algebra to the new class. An analogue of this new class in module theory is related to the broader problem of determining when a lattice of submodules forms a specific algebraic structure. We establish the necessary and sufficient conditions for a lattice of submodules of a module to admit such an algebraic structure.