On Jacobson type radicals of near-rings

The main result of this paper is a negative answer to the problem concerning the radicalness in the sense of Kurosh—Amitsur of the near-ring radicalJ ₀. We shall give negative answers to some hereditariness problems of near-ring radicals too.     

论域上和可换环上的群代数的Jacobson根基

本文讨论charK=p≠0之域K上的有限群群代数K[G]的Jacobson根基,推广Bedl关于Frobenius群群代数之J—根基的结果,并讨论特征为P~t的可换环上的群环的J—根基。本文记法同[1]。 §Ⅰ特征为p≠0的域K上有限群群代数的根基 Maschke定理指出,若ο(G)<∞,则JK[G]=0当且仅当chark=0或charK=p且p(G)。对于charK=p且p|o(G)的情况[2]指出:若G是有补P的Frobenius群,P是G的Sylow p—子群,则JK[G]=∩JK[P~x]K[G]。对于满足上述条件的K[G],x∈G     

The Jacobson radical of the Laurent series ring

For a large class of rings A, including all rings with right Krull dimension, it is proved that for every automorphism ϕ of the ring A, the Jacobson radical of the skew Laurent series ring A((x, ϕ)) is nilpotent and coincides with N((x, ϕ)), where N is the prime radical of the ring A. If A/N is a ring of bounded index, then the Jacobson radical of the Laurent series ring A((x)) coincides with N((x)). __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 2, pp. 209–215, 2006.     

On Ext-indices of ring extensions

In this paper, we investigate the finiteness of Ext-indices for certain ring extensions. In this direction, we introduce some conjectures and discuss the relationships among them. We also prove these conjectures in some special cases. Furthermore, we prove that the trivial extension of an Artinian local ring by its residue class field is always of finite Ext-index, and prove a generalization of the Auslander-Reiten conjecture for this type of ring.     

Integral ring extensions and lo-pairs

We give an example of an LO-pair (R,T) with dim R=dimT = n, so that all maximal ideals of R have the same height n, but with T not integral over R. This answers an open question of David E. Dobbs.     

Minimal silting modules and ring extensions

Ring epimorphisms often induce silting modules and cosilting modules, termed minimal silting or minimal cosilting. The aim of this paper is twofold. Firstly, we determine the minimal tilting and minimal cotilting modules over a tame hereditary algebra. In particular, we show that a large cotilting module is minimal if and only if it has an adic module as a direct summand. Secondly, we discuss the behavior of minimality under ring extensions. We show that minimal cosilting modules over a commutat...     

LCM-splitting sets in some ring extensions

Let be a saturated multiplicative set of an integral domain . Call an lcm splitting set if and are principal ideals for every and . We show that if is an -stable overring of (that is, if whenever and is principal, it follows that and if is an lcm splitting set of , then the saturation of in is an lcm splitting set in . Consequently, if is Noetherian and is a (nonzero) prime element, then is also a prime element of the integral closure of . Also, if is Noetherian, is generated by prime elements of and if the integral closure of is a UFD, then so is the integral closure of .