A note on locally nilpotent rings with chain conditions

It is shown that a locally nilpotent ring with maximality condition for two-sided ideals is nilpotent. The restriction on the characteristic in one of the author's previously published theorems is lifted. A one-sided nil-ideal of an alternative ring, satisfying the maximality condition for right ideals, is a nilpotent ring. An example is constructed of a commutative locally nilpotent ring A with maximality condition for ideals which is idempotent: A = A².The article was prepared for print by the author and submitted after his death which took place on February 24, 1972.Translated from Matematicheskie Zametki, Vol. 12, No. 2, pp. 121–126, August, 1972.     

Semistar ascending chain conditions over polynomial rings

Ricerche di Matematica - Let D be an integral domain. G. Picozza associated to a stable semistar operation $$star $$ on D,  a semistar operation $$star _1$$ on the polynomial ring D[X]....     

Semistar ascending chain conditions over power series rings

Ricerche di Matematica - Let D be an integral domain. We associate to a semistar operation $$\star $$ on D, a semistar operation $$*$$ on D[[X]]. We prove that if D satisfies the $$\star _f$$...     

Near rings associated with Sperner spaces

In this paper we introduce Sperner spaces with operators, denoted by TSO, and associate to each such space a near-ring. The associated near-ring provides a generalization of the kernel of an affine translation space. We obtain a characterization of those TSO for which the associated near-ring is a near-field. Special attention is given to finite TSO's with cyclic monoids of operators.     

The ar property and chain conditions in group rings

It is shown that ifJ is the ring of integers or a field andG a group such that the augmentation ideal of the group ringJG has the AR property then the ringJG and the groupG satisfy certain chain conditions.     

Near rings without zero divisors

Near rings without zero divisors, and a dual structure, near codomains, are studied. It is shown that a near ring is a near field if and only if it is an integral near ring, a near codomain, and has a non-zero distributive element. If the additive group (N, +) of a near integral domainN is cohopfian, then (N, +) possesses a fixed point free automorphism which is either torsion free or of prime order. This generalizes a well-known theorem of Ligh for finite near integral domains. A result ofGanesan [1] on the non-zero divisors in a finite ring is generalized to near rings.     

Semilocal rings with Engel conditions

The relation between the Engel structure of a semilocal ring and that of its multiplicative group is investigated. Suppose that every local ring whose multiplicative group satisfies an m-Engel condition for some positive integer m is an f (m)-Engel ring for some function f . It is proved that under this condition a corresponding statement holds for every semilocal ring which is generated by its multiplicative group. Received: 20 September 2005     

More on the AR property and chain conditions in group rings

We investigate when the augmentation ideal of a group algebrakG has the AR property in the cases ofG a locally nilpotent or hyperabelian group.