Polynomial Rings and Totally Ordered Monoid Rings

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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 G-Armendarizring 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 G-Armendariz ring R is Baer then RG is Baer. We also introduced the concept of a G-quasi-Armendariz 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 G-quasi-Armendariz for any totally ordered monoid G. From this we know that any quasi-Baer ring is G-quasrArmendariz. Hence we proved that if R is' quasi-Baer then RG is quasi-Baer.2. Let I be an ideal I of a ring R. We considered when the annihilator of I in any left R-module M is a direct summaud of … More M. In other words, we considered when the preradical which assigns for any left R-module M the annihilator of I in M, is splitting. We showed that if an ideal I satisfies this condition and if R is I-torsion-free, 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 D-simple for a derivation D. Less