New Additive Mappings On G-Rings With Involution
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Date
2017-08
Authors
Kadhim, Ali Kareem
Journal Title
Journal ISSN
Volume Title
Publisher
Universiti Sains Malaysia
Abstract
One of the most important problems in G-ring theory is to determine when a G-ring is
commutative. In this study, the problem is extended to a new algebraic structure G-rings with
involution (also known as G∗-rings). This research began at the point when researchers were
starting to pay attention to G-rings with involution as having a similar feature as that of rings
with involution. G-ring was first introduced in 1964 to obtain some generalization on the theory
of rings. A G-ring involves two additive abelian groups, one of which is G, and the other
labeled as M for which there is an additive mapping M ×G×M → M satisfying the usual
distributive and associative properties as in rings. In 1966, a weaker version of G-ring was
created by Barnes to have some further generalizations. The work done in this research was
based on the definition by Barnes. Relevant results on rings and rings with involution (often
referred to as ∗-rings) were studied to obtain significant ideas on the work that can be done in
G-rings with involution. Studies on rings with involution have largely motivated this research
since the established results can be applied in G-rings with involution. Some significant results
on G-rings are also applied to produce the results for G-rings with involution. This thesis
focuses only on prime and semiprime G-rings with involution. In investigating the commutative
property of prime and semiprime G-rings with involution, similar to rings and ∗-rings, new
additive mappings are created which are the Jordan G∗-centralizers, reverse G∗-centralizers,
G∗-derivations, Jordan G∗-derivations, G∗-derivation pairs and Jordan G∗-derivation pairs. In
addition to discovering the commutative property of a G-ring M with involution, the relations
between these new additive mappings defined on M are also studied. In this thesis, it was
proven that although every reverse G∗-centralizer is a Jordan G∗-centralizer, the converse is
true only if M is semiprime and 2-torsion free. In many of the results presented in this thesis,
a flexibility condition called Assumption (A) is used. Thus, for the case of M above, it was
proven that any additive mapping T on M is a reverse G∗-centralizer provided that T satisfies
one of two equations. Some other results which utilizes Assumption (A) include establishing
when a G∗-derivation is zero if M is semiprime, when M is commutative if M is prime or
semiprime, and when a Jordan G∗-derivation pair is a G∗-derivation pair if M is 6-torsion free.
The center of M, Z(M), also plays a role in the commutative property of M. In particular,
if all commutators of skew-hermitian elements is in Z(M), M is proven to be a commutative
G-ring with involution (using Assumption (A)). Other results involving Z(M) include the case
when a 2-torsion free, semiprime G-ring M is normal, which yields that all Hermitian elements
are contained in Z(M). For the case when M is 2-torsion free, prime and non-commutative
but normal, it yields that all elements of Z(M) are Hermitian elements. Besides the results
mentioned above, several other properties of the additive mappings on M are also established
in this thesis.
Description
Keywords
New additive mappings , on g-rings with Involution