Throughout, R is a commutative ring with identity and X is an indeterminate over R. We consider R{X}, the polynomial ring in one indeterminate over R, and G(R), the group of R-automorphisms of R{X}. In particular, we consider the subring of R{X} left fixed by the group G(R), denoted by R{X}('G(R)). Let B(R) be the subgroup of G(R) such that (sigma) (ELEM) B(R) if and only if (sigma)(X) = a + bX, b a unit of R. If R is reduced, then G(R) = B(R); otherwise, B(R) (L-HOOK) G(R). We prove in Chapter I that R{X}('G(R)) = R{X}('B(R)). / In Chapter I we also prove that for R to be properly contained in R{X}('G(R)), it is necessary that R/M is a finite field for some maximal ideal M of R. Hence, if R is a quasi-local ring with maximal ideal M and R/M is infinite, then R{X}('G(R)) = R. / Let R be a quasi-local ring with maximal ideal M such that R/M is isomorphic to the Galois field with p('s) elements, where p is a prime integer and s (ELEM) Z('+). In Chapter II, we show that / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) / where / (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) / In particular, we determine Z(,n){X}('G(Zn)) for n (ELEM) Z('+). Moreover, we prove that R{X}('G(R)) contains a nonconstant monic polynomial if and only if R is a 0-dimensional SFT-ring. / In Chapter III, we investigate R{X}('G(R)) for a von Neumann regular ring R. We obtain equivalent conditions for R{X}('G(R)) to contain a nonconstant monic polynomial; one of these is that {card(R/M)} is bounded for all maximal ideals M of R. Moreover, we prove that R is properly contained in R{X}('G(R)) if and only if R has a direct summand S such that S{X}('G(S)) contains a nonconstant monic polynomial. Finally, in Chapter III we construct a von Neumann regular ring B such that B/M is finite for infinitely many maximal ideals M of B, but B{X}('G(B)) = B. / In Chapter IV, we show that for any commutative ring R with identity, R{X}('G(R)) contains a nonconstant monic polynomial if and only if R is 0-dimensional, card(R/M) < N for some N (ELEM) Z('+) and for all maximal ideals M of R, and nilpotent elements have bounded order of nilpotency. / Source: Dissertation Abstracts International, Volume: 43-03, Section: B, page: 0747. / Thesis (Ph.D.)--The Florida State University, 1982.
| Identifer | oai:union.ndltd.org:fsu.edu/oai:fsu.digital.flvc.org:fsu_74802 |
| Contributors | DOWLEN, MARY MARGARET., Florida State University |
| Source Sets | Florida State University |
| Detected Language | English |
| Type | Text |
| Format | 54 p. |
| Rights | On campus use only. |
| Relation | Dissertation Abstracts International |
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