Одно замечание о периодических кольцах

We obtain a new and non-trivial characterization of periodic rings (that are those rings $R$ for which, for each element $x$ in $R$, there exists two different integers $m$, $n$ strictly greater than $1$ with the property $x^m=x^n$) in terms of nilpotent elements which supplies recent results in this subject by Cui--Danchev published in (J. Algebra \& Appl., 2020) and by Abyzov--Tapkin published in (J. Algebra \& Appl., 2022). Concretely, we state and prove the slightly surprising fact that an arbitrary ring $R$ is periodic if, and only if, for every element~$x$ from $R$, there are integers $m>1$ and $n>1$ with $m\not= n$ such that the difference $x^m-x^n$ is a nilpotent.

On the idempotent and nilpotent sum numbers of matrices over certain indecomposable rings and related concepts

We investigate a few special decompositions in arbitrary rings and matrix rings over indecomposable rings into nilpotent and idempotent elements. Moreover, we also define and study the nilpotent sum trace number of nilpotent matrices over an arbitrary ring. Some related notions are explored as well.

A new algorithm for computing idempotents of ℛ-trivial monoids

The authors of Berg et al. [J. Algebra 348 (2011) 446–461] provide an algorithm for finding a complete system of primitive orthogonal idempotents for [Formula: see text], where [Formula: see text] is any finite [Formula: see text]-trivial monoid. Their method relies on a technical result stating that [Formula: see text]-trivial monoid are equivalent to so-called weakly ordered monoids. We provide an alternative algorithm, based only on the simple observation that an [Formula: see text]-trivial monoid may be realized by upper triangular matrices. This approach is inspired by results in the field of coupled cell network dynamical systems, where [Formula: see text]-trivial monoids (the opposite notion) correspond to so-called feed-forward networks. We first show that our algorithm works for [Formula: see text], after which we prove that it also works for [Formula: see text] where [Formula: see text] is an arbitrary ring with a known complete system of primitive orthogonal idempotents. In particular, our algorithm works if [Formula: see text] is any field. In this respect our result constitutes a considerable generalization of the results in Berg et al. [J. Algebra 348 (2011) 446–461]. Moreover, the system of idempotents for [Formula: see text] is obtained from the one our algorithm yields for [Formula: see text] in a straightforward manner. In other words, for any finite [Formula: see text]-trivial monoid [Formula: see text] our algorithm only has to be performed for [Formula: see text], after which a system of idempotents follows for any ring with a given system of idempotents.

The set of strong torsion elements of a module over a noncommutative ring

Let [Formula: see text] be an arbitrary ring and [Formula: see text] be a nonzero right [Formula: see text]-module. In this paper, we introduce the set [Formula: see text], for some nonzero ideal [Formula: see text] of [Formula: see text] of strong torsion elements of [Formula: see text] and the properties of this set are investigated. In particular, we are interested when [Formula: see text] is a submodule of [Formula: see text] and when it is a union of prime submodules of [Formula: see text].

Almost prime and weakly prime submodules

Let [Formula: see text] be an arbitrary ring and [Formula: see text] be a right [Formula: see text]-module. A proper submodule [Formula: see text] of [Formula: see text] is called almost prime (respectively, weakly prime) if for each submodule [Formula: see text] of [Formula: see text] and each ideal [Formula: see text] of [Formula: see text] that [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]), then [Formula: see text] or [Formula: see text]. We study these notions which are new generalizations of the prime submodules over noncommutative rings and we obtain some related results. We show that these two concepts in some classes of modules coincide. Moreover, we investigate the conditions that [Formula: see text] is almost prime, where [Formula: see text] is a submodule of [Formula: see text] and [Formula: see text] is an ideal of [Formula: see text]. Also, the almost prime radical of modules will be introduced and we extend some known results.

Balance for relative cohomology of complexes

Let [Formula: see text] be an arbitrary ring. We use a strict [Formula: see text]-resolution [Formula: see text] of a complex [Formula: see text] with finite [Formula: see text]-projective dimension, where [Formula: see text] denotes a subcategory of right [Formula: see text]-modules closed under extensions and direct summands and admits an injective cogenerator [Formula: see text], to define the [Formula: see text]th relative cohomology functor [Formula: see text] as [Formula: see text]. If a complex [Formula: see text] has finite [Formula: see text]-injective dimension, then one can use a dual argument to define a notion of a relative cohomology functor [Formula: see text], where [Formula: see text] is a subcategory of right [Formula: see text]-modules closed under extensions and direct summands and admits a projective generator. Under several orthogonal conditions, we show that there exists an isomorphism [Formula: see text] of relative cohomology groups for each [Formula: see text]. This result simultaneously encompasses a balance result of Holm on Gorenstein projective and injective modules, a balance result of Sather-Wagstaff, Sharif and White on Gorenstein projective and injective modules with respect to semidualizing modules, and a balance result of Liu on Gorenstein projective and injective complexes. In particular, as an application of this result, we extend the above balance result of Sather-Wagstaff, Sharif and White to the setting of complexes.

A note on the FIP property for extensions of commutative rings

A ring extension R ? S is said to be FIP if it has only finitely many intermediate rings between R and S. The main purpose of this paper is to characterize the FIP property for a ring extension, where R is not (necessarily) an integral domain and S may not be an integral domain. Precisely, we establish a generalization of the classical Primitive Element Theorem for an arbitrary ring extension. Also, various sufficient and necessary conditions are given for a ring extension to have or not to have FIP, where S = R[?] with ? a nilpotent element of S.

A dual approach to the theory of inverse split modules

Let [Formula: see text] be an arbitrary ring with identity, [Formula: see text] a right [Formula: see text]-module and [Formula: see text] a fully invariant submodule of [Formula: see text]. The notion of an [Formula: see text]-inverse split module [Formula: see text] has been defined and studied by the present authors recently. In this paper, we introduce its dual notion, namely, dual [Formula: see text]-inverse split module [Formula: see text]. This work is devoted to investigation of various properties and characterizations of a dual [Formula: see text]-inverse split module [Formula: see text]. We include applications for rings and cosingular submodules. We also deal with the notion of relatively dual inverse splitness to investigate direct sums of dual inverse split modules.