Uniformly Primal Submodule over Noncommutative Ring

Let R be an associative ring with identity and M be a unitary right R-module. A submodule N of M is called a uniformly primal submodule provided that the subset B of R is uniformly not right prime to N , if there exists an element s ∈ M − N with sRB ⊆ N .The set adj N = r ∈ R | mRr ⊆ N for some m ∈ M is uniformly not prime to N .This paper is concerned with the properties of uniformly primal submodules. Also, we generalize the prime avoidance theorem for modules over noncommutative rings to the uniformly primal avoidance theorem for modules.

A Survey on Some Algebraic Characterizations of Hilbert’s Nullstellensatz for Non-commutative Rings of Polynomial Type

In this paper we present a survey of some algebraic characterizations of Hilbert’s Nullstellensatz for non-commutative rings of polynomial type. Using several results established in the literature, we obtain a version of this theorem for the skew Poincaré-Birkhoff-Witt extensions. Once this is done, we illustrate the Nullstellensatz with examples appearing in noncommutative ring theory and non-commutative algebraic geometry.

Reflexive property on rings with involution

Mason introduced the notion of reflexive property for rings which play roles in noncommutative ring theory. In this paper, we extend this property to rings with involution and investigate their properties. We provide many examples of these rings and also consider some extensions such as trivial extension, Dorroh extension, etc.

A Characterization of Derivations in Prime Rings with Involution

The purpose of this paper is to investigate $*$-differential identities satisfied by pair of derivations on prime rings with involution. In particular, we prove that if a 2-torsion free noncommutative ring $R$ admit nonzero derivations $d_1, d_2$ such that $[d_1(x), d_2(x^*)]=0$ for all $x\in R$, then $d_1=\lambda d_2$, where $\lambda\in C$. Finally, we provide an example to show that the condition imposed in the hypothesis of our results are necessary.

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].

Structure of insertion property by powers

This paper, concerns a class of rings which satisfies the Abelian property in relation to the insertion property at zero by powers and local finite. The concepts of Insertion of-Power-Factors-Property (PFP) and principal finite are introduced for the purpose, and the structures of IPFP, Abelian and locally (principally) finite rings are investigated in relation with several situations of matrix rings and polynomial rings. Moreover, the results obtained here are widely applied to various sorts of rings which have roles in the noncommutative ring theory.

Insertion of units at zero products

The purpose of this paper is to provide useful connections between units and zero divisors, by investigating the structure of a class of rings in which Köthe’s conjecture (i.e. the sum of two nil left ideals is nil) holds. We introduce the concept of unit-IFP for the purpose, in relation with the inserting property of units at zero products. We first study the relation between unit-IFP rings and related ring properties in a kind of matrix rings which has roles in noncommutative ring theory. The Jacobson radical of the polynomial ring over a unit-IFP ring is shown to be nil. We also provide equivalent conditions to the commutativity via the unit-IFP of such matrix rings. We construct examples and counterexamples which are necessary to the naturally raised questions.