On the structure of polynomial near-rings /
Given a zero-symmetric right near-ring N with identity, it is known that the standard set of polynomials over N with the usual rules for polynomial addition and multiplication is not in general a near-ring. This indicates the need for an alternate definition for N[x]. Let K denote the natural number...
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| Format: | Thesis Book |
| Language: | English |
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[Place of publication not identified] :
[publisher not identified] ;
1999.
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| Online Access: | http://proxy.library.tamu.edu/login?url=http://proquest.umi.com/pqdweb?did=731681141&sid=1&Fmt=2&clientId=2945&RQT=309&VName=PQD |
| Summary: | Given a zero-symmetric right near-ring N with identity, it is known that the standard set of polynomials over N with the usual rules for polynomial addition and multiplication is not in general a near-ring. This indicates the need for an alternate definition for N[x]. Let K denote the natural numbers and regard functions of N[] = {f : K -> N} as sequences in N. For each a [] N let L[] : N[] -> N[] be the map given by L[](n[], n[],Ö) = (0, n[], n[],Ö) and let x : N[] ̉N[] be the map given by x(n[], n[],Ö) = (0, n[], n[],Ö). Observe that M(N[]) = {f : N[] -> N[]} is a near-ring under function composition and pointwise addition. Then N[x] is defined to be the subnear-ring of M(N[]) generated by {x}[]{L[]| a [] N}. The above definition for N(x) was first given by S. Bagley in his 1993 doctoral dissertation, in which he proved that N(x) is isomorphic to the standard polynomial ring in one indeterminate in the case that N is a ring. Bagley also introduced two maps, (.)+ and (.)*, from the set of ideals of N to the set of ideals of N(x). For an ideal I of a ring N both I * and I+ give rise to the set of polynomials with coefficients from I , but in general these ideals may differ. We further investigate these maps, providing an inverse for (.)* in the process. We then introduce new ideal mappings to show that {I*| I is an ideal of N} is in one-to-one correspondence with the set of ideals of N[x] containing x. Bagley also gave a partial representation theorem for elements of N[x] in terms of an ideal sym₀ of N(x) . We provide a characterization of this ideal and use it to study the quotient structure N[x]/sym₀. In particular, we prove that N[x]/sym₀ inherits the properties of simplicity, planarity, and integrality from N only when N equals its distributor ideal, or equivalently when sym₀ is the ideal generated by x. We also introduce a noetherian quotient-like structure to obtain a left ideal (L[] : α) in N(x) from a left ideal L of N and any α [] N[]. We prove a theorem about the non-maximality of (L[] : α) and, for restricted N and or, we give necessary and sufficient conditions for (L[] : α) to be a two-sided ideal of N(x). We then define and study more general versions of polynomial near-rings. Finally, we consider the distributive and central elements of polynomial near-rings. We define a new centralizer near-ring M[](N) for each subset S of N[], the distributive elements of N. For any ring N, we show that M[](N[x]) = C(N[x]), the center of N(x) . However, we prove that there exist non-rings N for which even the near-ring generated by C(N[x]) is properly contained in M[](N[x]). |
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| Item Description: | Vita. "Major Subject: Mathematics". |
| Physical Description: | vi, 60 leaves ; 28 cm. Issued also on microfiche from University Microfilm Inc. |
| Bibliography: | Includes bibliographical references (leaves 58-59). |