Polynomial Rings and Totally Ordered Monoid Rings

1.作为多项式环的推广，我们考虑了一个环和一个全序G的么半群环RG，如果RG的任意两个元素的乘积为零意味着它们的所有系数的乘积为零，则称R为G-Armendarizing环。我们证明了这个条件等价于R的左零化子理想集与RG的左零化子理想集之间的自然映射的双射化。由此我们知道，如果G-Armendariz环R是Baer，则RG是Baer。我们还引入了G-拟Armendariz环的概念，并给出了类似的刻划。证明了如果R的任一主左理想的左零化子是纯左理想，则R对任一全序么半群G是G-拟Armendariz环，由此我们知道任何拟Baer环都是G-拟Armendariz环。从而证明了如果R是‘拟Baer的，则RG是拟Baer的。设I是环R的理想I，我们考虑了当I的零化子在任意左R-模M中是…的直和时换句话说，我们考虑了赋给任何左R-模M的I在M中的零化子的预根是分裂的。证明了：如果理想I满足这个条件，且R是I-挠自由的，则对任何包含I的理想H，R/H都是右遗传右完全环。特别地，当R是可交换的时，我们给出了理想I具有这种性质的充要条件。此外，作为Osofsky和Smith的一个结果的应用，我们证明了：如果环R的所有非零理想I都具有这一性质，则R的任何非零Fector环都是素环的直和。设R是环，U(R)表示R中的单位群，我们用通常的teft乘法把R看作是左U(R)模。我们证明了轨道数是有限的当且仅当R是一个有限环和无限多个环的直和。我们还证明了如果R没有非零的有限Fector环，则这个条件等价于R是左Artin左分配环。1981年D。A.Jordan证明了没有可逆导数的微分环的存在性。与此相关，我们证明了在一定条件下，具有n个变量和n个交换导子的斜多项式环对于导子D是D-单的。

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