| Summary: | In this thesis the ideal structure of near-rings and the associated matrix
near-rings is studied. Firstly, two aspects of near-rings are investigated, these
are: the decomposition of a near-ring; and the nilpotency of the Jacobson
s-radical, via the nil-rigid series. Secondly, the interplay of ideals of a nearring
and the ideals of the associated matrix near-ring is investigated.
The socle ideal (an anti-radical) of a near-ring R is characterized as an intersection
of annihilators of R-groups of a special type, called type-K. That
is, the socle ideal is an annihilator ideal.
The R-groups of type-K are shown to separate the Jacobson 0-radical and
the Jacobson s-radical, if there is at least one R-group of type-K contained
in some type-0 R-group.
Alternating chains of R-groups of type-0 and type-K are defined and will be
called
-chains. It is then proved that the maximal length of the
-chains
give a lower bound on the nil-rigid length of the near-ring. Consequently,
one can directly determine that the nil-rigid length of the matrix near-ring,
Mn(R), is equal or bigger than the nil-rigid length of R from the structure
of any one of the finite faithful R-groups of R.
For an ideal I being a Jacobson type radical, socle ideal or s-socle ideal
of a near-ring, open-questions on the relationship between I+, I#3; and the
matrix near-ring ideals of the same type are answered.
A systematic theory is developed that connects each annihilator ideal I
of R to its corresponding matrix near-ring ideal, I. This is made possible by
the use of Action 2 of the matrix near-ring on the Mn(R)-groups. Specific
conditions are given which ensure that I+ #18; I and I #18; I#3;. Examples
illustrate cases in which no such relationships exist. Necessary and sufficient
conditions are given for I+ #18; I #18; I#3; to hold.
This thesis thus illustrates the relationship between the structure of a faithful
R-group and the ideal structure of the associated matrix near-rings.
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