Structural and combinatorial aspects of graded rings with applications
Creators: name: Plant, A
Publisher: name: University Of Tasmania
Year: 2023

description: Algebraic structure is at the heart of mathematics and graded ring structures arise in many natural applications and contexts. Particular examples of graded rings actively investigated in recent decades include generalized matrix rings, the Morita rings associated with Morita contexts, polynomial rings and the ring of symmetric functions. We describe a graded construction for all these rings. In order to extend our investigation to as wide a class of graded rings as possible, we consider rings graded by partial groupoids. We present homogeneous sums as graded by induced partial groupoids. Homogeneity of ideals and radicals of graded rings is one the most interesting and fundamental ideas in graded ring theory. We introduce a consistent definition for the graded Jacobson radical for group graded rings without unity. We compare the graded Jacobson radical for both rings with unity, and those without. We find that for group graded rings, the descriptions are equivalent. We provide some necessary lemmas for rings without unity which have appeared in the case the ring is afforded unity. These lay the relevant foundation for our investigations. For example, we show that J(R) ‚Äöv†¬© R\\(_e\\) ‚Äöv§vú J(R\\(_e\\)) (where e is idempotent) for all groupoid graded rings without unity. We give a generalization of Bergman's 'folklore' lemma for group graded rings with unity to partial groupoid graded rings without unity. Since homogeneous sums and generalized matrix rings are both graded by induced partial groupoids, our generalization of Bergman's lemma applies to these graded rings as well. Our results also yield three corollaries on the Jacobson radical of graded F-algebras (where F is a field). In 1985 Anderson, Divinsky, and Sulinski defined an invariant ideal Io in which R\\(_1\\)I\\(_0\\)R\\(_1\\) ‚Äöv§vú I\\(_0\\) for any Z\\(_2\\)-graded ring R and found that the Jacobson radical was 'invariant' in all Z\\(_2\\)-graded rings. We define a new concept of S-invariance and it turns out that the results of several previous authors fit our definition. For example a 1989 result of Jespers and Wauters is equivalent to saying that the Jacobson radical is S-invariant in all generalized matrix rings. We specify, with necessary and sufficient conditions on S, exactly for which graded rings the Jacobson radical is S-invariant. The ring of symmetric functions is a graded ring with important applications in mathematical physics. Structural aspects of this graded ring are described. Using the transition matrices of the symmetric functions we are able to write the spin characters of the symmetric group in terms of the ordinary ones. This leads us to describe a new algorithm for the spin characters. We also present simpler algorithms in two special cases. We include the ring of Hirota derivatives as a practical example of a graded ring without unity. The BKP equations are one example of its homogeneous elements. Setting up this example leads us to introduce the generalized Q-operators and we describe some connections between them and the BKP equations. By associating the generalized Q-operators with shifted Young diagrams, we generate the lower weight portion of the BKP hierarchy. Motivation for these studies is directed mostly by the investigations of other authors and driven mostly by curiosity.