1994 Fiscal Year Final Research Report Summary
Considerations on the unit group of a ring and unramified extensions
| Project/Area Number |
05640033
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| Research Category |
Grant-in-Aid for General Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Research Field |
Algebra
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| Research Institution | Aichi University of Education |
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Principal Investigator |
KANEMITSU Mitsuo Aichi University of Education, Faculty of Education, Professor, 教育学部, 教授 (60024014)
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| Co-Investigator(Kenkyū-buntansha) |
YOSHIDA Kenichi Okayama University of Science, Faculty of Science, Professor, 理学部, 教授 (60028264)
SUZUKI Masashi Assitant Professor, 教育学部, 助教授 (50216438)
ISHITOYA Kiminao Assistant Professor, 教育学部, 助教授 (80133130)
OHTA Minoru Professor, 教育学部, 教授 (30022635)
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| Project Period (FY) |
1993 – 1994
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| Keywords | Anti-integral extension / The unit group of a ring / Flat extension / Differential module / Unramified extension |
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| Research Abstract |
All rings are assumed to be commutative with identity. Let R be a Noetherian integral domain with quotient field K and let alpha be a non-zero element of an algebraic field extension L of K.Also, let pi : R[X] * R[alpha] be the R-algebra homomorphism sending X to alpha and letpsi_<alpha>(X)=X^d+eta_1X^<d-1>+・・・+eta_d be the monic minimal polynomial of alpha over K.Now, put I_<beta>={r*R|rbeta*R}for beta*L,put I_<[alpha]>=I_<eta1>*I_<eta2>*・・・*I_<etad> and put J_<[alpha]>=I_<alpha>c (psi_<alpha> (X)) where c (psi_<alpha> (X)) denotes the R-submodule of K generated by the coefficients of psi_<alpha>(X). Moreover, put I^^-_<[alpha]>=I_<[alpha]>(1, eta_1, ・・・, eta_<d-1>) and put J^^-_<[alpha]>=eta_dI_<[alpha]>. The element alpha is called an anti-integral element of degree d over R if Ker pi=I_<[alpha]>psi_<alpha>(X)R[X]. When alpha is an anti-integral element over R,A=R_<[alpha]> is called an anti-integral extension of R.In case L=K,it its well known that R[alpha] is an anti-integral extension of R if and only if R=R[alpha]*R[alpha^<-1>].
Assume that A=R[alpha] is an anti-integral extension of R.Then we obtain the following results :
1. I^^-_<alpha>=R if and only if the restriction mapping psi : Spec(A) * Spec(R) is surjective and A is a flat R-module.
2. A is an unramified extension over R,that is, the differential module OMEGA_R(A)=(0) if and only if I_<[alpha]>psi'_<alpha>(alpha)A=A.In this case, A is a flat R-module. This means that unramified extensions relate the unit elements.
3. A is a flat R-module if and only if I^^-_<[alpha]>A=A.In case L=K,it is equivalent that A is an unramified extension of R.
4. Let alpha_i(1<less than or equal>i<less than or equal>n)be elements of L such that the R[alpha_i] are unramified over R.Then R[alpha_1, alpha_2, ・・・, alpha_n] is an unramified extention over R.
5.If A is integral over R,then alpha is a unit of A and only if eta_d is a unit of R.
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