Polynomial Rings and Totally Ordered Monoid Rings
Project/Area Number 
13640025

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  OKAYAMA UNIVERSITY 
Principal Investigator 
HIRANO Yasuyuki Okayama University, Faculty of Science, Associate Professor, 理学部, 助教授 (90144732)

CoInvestigator(Kenkyūbuntansha) 
NAKAJIMA Atsusi Okayama University, Faculty of Environmental Science and Technology, Professor, 環境理工学部, 教授 (30032824)
IKEHATA Shuichi Okayama University, Faculty of Environmental Science and Technology, Professor, 環境理工学部, 教授 (20116429)

Project Period (FY) 
2001 – 2002

Project Status 
Completed (Fiscal Year 2002)

Budget Amount *help 
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2002: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2001: ¥1,200,000 (Direct Cost: ¥1,200,000)

Keywords  ring / module / totally ordered monoid / polynomial ring / monoid ring / annihilator ideal / distributive ring / differential operator / モノイド / 順序半群 / 零下イデアル / 平坦 
Research Abstract 
1. As a generalization of a polynomial ring, we considered the monoid ring RG for a ring and a totally ordered G. Aring R is called a GArmendarizring if the product of any two elements of RG is zero implies that the products of all of their coefficients are zero. We proved that this condition is equivalent to the bijectivify of the natural mapping between the set of left annihilator ideals of R and the set of that of RG. From this we know that if a GArmendariz ring R is Baer then RG is Baer. We also introduced the concept of a GquasiArmendariz ring and gave a similar charactrization. We showed that if the left annihilator of any principal left ideal of R is a pure left ideal then R is GquasiArmendariz for any totally ordered monoid G. From this we know that any quasiBaer ring is GquasrArmendariz. Hence we proved that if R is' quasiBaer then RG is quasiBaer. 2. Let I be an ideal I of a ring R. We considered when the annihilator of I in any left Rmodule M is a direct summaud of
… More
M. In other words, we considered when the preradical which assigns for any left Rmodule M the annihilator of I in M, is splitting. We showed that if an ideal I satisfies this condition and if R is Itorsionfree, then, for any ideal H containing I, R/H is a right hereditary right perfect ring. In particular, when R is commutative, we gave a necessary and sufficient condition for an ideal I to have this property. Moreover, as an application of a result of Osofsky and Smith, we proved that if all nonzero ideal I of a ring R have this property then any nonzero fector ring of R is a direct sum of prime rings. 3. Let R be a ring and let U(R) denote the group of units in R. We consider R as a left U(R)moduIe by the usual teft multiplication. We proved that the number of orbits is finite if and only if R is the direct sum of a finite ring and fuiitely many muserial rings. We also proved that if R has no nonzero finite fector ring, then this condition is equivalent to that R is a left Artinian left distributive ring. 4. In 1981D. A. Jordan has shown the exsistance of a differential ring with no invertible derivation. In connection with this, we showed that under certain condition, a skew polynomial ring with n variables and n commutative derivations is Dsimple for a derivation D. Less

Report
(3 results)
Research Products
(12 results)