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Structural and combinatorial aspects of graded rings with applications

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Plant, Allison 2008 , 'Structural and combinatorial aspects of graded rings with applications', PhD thesis, University of Tasmania.

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Abstract

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 def-
inition 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) ∩ Re ⊆ J(Re) (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
R1IoR1 ⊆ Io for any Z2-graded ring R and found that the Jacobson radical
was 'invariant' in all Z2-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 equa-
tions. 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.

Item Type: Thesis - PhD
Authors/Creators:Plant, Allison
Keywords: Graded rings, Rings (Algebra)
Copyright Holders: The Author
Copyright Information:

Copyright 2008 the author - The University is continuing to endeavour to trace the copyright
owner(s) and in the meantime this item has been reproduced here in good faith. We
would be pleased to hear from the copyright owner(s).

Additional Information:

Available for library use only and copying in accordance with the Copyright Act 1968, as amended. Thesis (PhD)--University of Tasmania, 2008. Includes bibliographical references. Introduction -- 1. Preliminaries -- 2. Graded rings -- 3. Rings with invariant radicals -- 4. Rings with homogeneous radicals -- 5. The ring of Hirota derivatives -- 6. Spin characters of the symmetric group

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