Estimate of the maximal cycle length in the graph of polynomial transformation of Galois–Eisenstein ring

The paper is concerned with polynomial transformations of a finite commutative local principal ideal of a ring (a finite commutative uniserial ring, a Galois–Eisenstein ring). It is shown that in the class of Galois–Eisenstein rings with equal cardinalities and nilpotency indexes over Galois rings there exist polynomial generators for which the period of the output sequence exceeds those of the output sequences of polynomial generators over other rings.

On noncommutative FGC rings

Many studies have been conducted to characterize commutative rings whose finitely generated modules are direct sums of cyclic modules (called FGC rings), however, the characterization of noncommutative FGC rings is still an open problem, even for duo rings. We study FGC rings in some special cases, it is shown that a local Noetherian ring R is FGC if and only if R is a principal ideal ring if and only if R is a uniserial ring, and if these assertions hold R is a duo ring. We characterize Noetherian duo FGC rings. In fact, it is shown that a duo ring R is a Noetherian left FGC ring if and only if R is a Noetherian right FGC ring, if and only if R is a principal ideal ring.

ABELIAN GROUPS WITH A p2-BOUNDED SUBGROUP, REVISITED

Let Λ be a commutative local uniserial ring of length n, p be a generator of the maximal ideal, and k be the radical factor field. The pairs (B, A) where B is a finitely generated Λ-module and A ⊆B a submodule of B such that pmA = 0 form the objects in the category [Formula: see text]. We show that in case m = 2 the categories [Formula: see text] are in fact quite similar to each other: If also Δ is a commutative local uniserial ring of length n and with radical factor field k, then the categories [Formula: see text] and [Formula: see text] are equivalent for certain nilpotent categorical ideals [Formula: see text] and [Formula: see text]. As an application, we recover the known classification of all pairs (B, A) where B is a finitely generated abelian group and A ⊆ B a subgroup of B which is p2-bounded for a given prime number p.

THE LATTICE OF PRERADICALS OVER LOCAL UNISERIAL RINGS

In this paper we describe the lattice of preradicals over any local uniserial ring (equivalently, any local artinian principal ideal ring). This lattice is isomorphic to a lattice of binary sequences of length n, where n is the length of the composition series for the ring, as a left and as a right module. We prove some of its properties: it is finite having cardinality 2n, it is distributive, self-dual and it is graded having rank [Formula: see text]. We also describe the correspondent posets of irreducible, join-irreducible, prime and coprime elements.

SOME MODEL THEORY OVER AN EXCEPTIONAL UNISERIAL RING AND DECOMPOSITIONS OF SERIAL MODULES

An example is given of a direct summand of a serial module that does not admit an indecomposable decomposition.

Nonsingular rings with essential socles

This paper is a study of nonsingular rings with essential socles. These rings were first investigated by Goldie [5] who studied the Artinian case and showed that an indecomposable nonsingular generalized uniserial ring is isomorphic to a full blocked triangular matrix ring over a sfield. The structure of nonsingular rings in which every ideal generated by a primitive idempotent is uniform was determined for the Artinian case by Gordon [6] and Colby and Rutter [2], and for the semiprimary case by Zaks [12]. Nonsingular rings with essential socles and finite identities were characterized by Gordon [7] and the author [10]. All these results were obtained by representing the rings in question as matrix rings. In this paper a matrix representation of arbitrary nonsingular rings with essential socles is found (section 2). The above results are special cases of this representation. A general method for representing rings as matrices is developed in section 1.