Near-rings of homogeneous maps /
Given an R-module RV, the centralizer near-ring determined by
| Main Author: | |
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| Format: | Thesis Book |
| Language: | English |
| Published: |
[Place of publication not identified] :
[publisher not identified] ;
1995.
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| Subjects: | |
| Online Access: | http://proxy.library.tamu.edu/login?url=http://proquest.umi.com/pqdweb?did=742745011&sid=1&Fmt=2&clientId=2945&RQT=309&VName=PQD |
| Summary: | Given an R-module RV, the centralizer near-ring determined by R and V is the set of functions MR(V) :=[] f : V[]V[] f(rv) = rf (v) for all v [] V and r [] R), endowed with pointwise addition and function composition as multiplication, and the elements are called homogeneous maps. Let C denote the collection of cyclic submodules of V. Then C determines a subnear-ring P[]R(V, C) of MR(V), where P[]R(V,C) := []f [] MR(V)[]f[]c extends to some [] [] EndR(V) for all C [] C[]. near-ring P[]R(V,C) is known as the near-ring of piecewise endomorphisms determined by V and C. Denote by PFn the domains R such that P[]R(Rn,C) = MR(Rn). Maxson and Van der Walt posed the problem in [MV2] of classifying PFn. We will show that for R Noetherian, R []PFn if and only if R is an one dimensional Gorenstein domain. For R a commutative Noetherian ring, we will determine the finitely generated modules V, for which MR(V) = EndR(V), and will provide a large class of examples for which MR(V) is a ring, but not the endomorphism ring. When MR(V) is commutative, MR(V) is certainly a ring. We will consider the converse for R a commutative ring. Examples will be provided to show that the converse is not true over commutative rings in general, although it will always be the case for uniform modules, modules over Artinian rings, modules over Dedekind domains, and for finitely generated modules over Noetherian rings. A module V for which it is not possible to change the addition without changing the action of R on V, will be defined to be a unique addition module. We will show that if R is commutative and Noetherian and V has a composition series and char(R/P)[] 2 for all maximal associated primes of V, then V is a unique addition module if and only if MR(V) = EndR(V). |
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| Item Description: | Vita. "Major Subject: Mathematics". |
| Physical Description: | vi, 65 leaves ; 28 cm. Issued also on microfiche from University Microfilms Inc. |
| Bibliography: | Includes bibliographical references. |