Semicommutative Rings Research Articles

One-sided nilpotent semicommutative properties of rings

In this paper, we study the semicommutative properties of rings with one-sided nilpotent structures. The concepts of left and right nilpotent semicommutative rings are defined and studied, which are a generalization of semicommutative rings. We show that the class of one-sided nilpotent semicommutative rings is strictly placed between the class of semicommutative rings and that of nil-semicommutative rings. As applications, we give some characterizations of semiprime rings from the point of view of left nilpotent semicommutative rings.

$$\Sigma$$-Semicommutative rings and their skew PBW extensions

$$\Sigma$$-Semicommutative rings and their skew PBW extensions

Left Nil Zero Semicommutative rings

This paper introduces a class of rings called left nil zero semicommutative rings ( LNZS rings ), wherein a ring R is said to be LNZS if the left annihilator of every nilpotent element of R is an ideal of R. It is observed that reduced rings are LNZS but not the other way around. So, this paper provides some conditions for an LNZS ring to be reduced and among other results, it is proved that R is reduced if and only if the ring of upper triangular matrices over R is LNZS. Furthermore, it is shown that the polynomial ring over an LNZS may not be LNZS and so is the case of the skew polynomial over an LNZS ring. Therefore, this paper investigates the LNZS property over the polynomial extension and skew polynomial extension of an LNZS ring with some additional conditions.

Semi-commutativity of graded rings and graded modules

A ring R is said to be semi-commutative if whenever a, b ∈ R such that ab = 0, then aRb = 0. In this article, we introduce the concepts of g−semi-commutative rings and g−N−semi-commutative rings and we introduce several results concerning these two concepts. Let R be a G-graded ring and g ∈ supp(R, G). Then R is said to be a g−semi-commutative if whenever a, b ∈ R with ab = 0, then aRgb = 0. Also, R is said to be a g − N−semi-commutative if for any a ∈ R and b ∈ N(R) ⋂ Ann(a), bRg ⊆ Ann(a). We introduce an example of a G-graded ring R which is g − N-semi-commutative for some g ∈ supp(R, G) but R itself is not semi-commutative. Clearly, if R is a g−semi-commutative ring, then R is a g − N−semi-commutative ring, however, we introduce an example showing that the converse is not true in general. Several results and examples are investigated. Also, we introduce the concept of g − NE−semi-commutative rings and we introduce several results concerning g−NE−semi-commutative rings. Let R be a G-graded ring and g ∈ supp(R, G). Then R is said to be a g−NE− semi-commutative ring if whenever a ∈ N(R) and b ∈ E(R) such that ab = 0, then aRgb = 0. Clearly, g−semi-commutative rings are g −NE−semi-commutative, however, we introduce an example ...

A Note on Generalized Semicommutative Rings

A Note on Generalized Semicommutative Rings

On Z-Symmetric Rings

Abstract We introduce the concept of Z-symmetric rings. In fact, the classes of all semicommutative rings, nil rings, reduced rings, Artinian rings and eversible rings are Z-symmetric rings. In order to sustain our assertion, we provide a number of examples of Z-symmetric and non Z-symmetric rings. We observe that the class of Z-symmetric rings lies strictly between the classes of eversible rings and the Dedekind finite rings. In particular, we consider the extensions of Z-symmetric rings. Finally, some new results between the Z-symmetric rings and Armendariz rings will be explored and investigated.

Semicommutative involution rings

In this paper, we establish a generalization of the class of semicommutative rings which is called * -semicommutative ring. We also discuss the relations among * -semicommutative rings, * -reversible rings and reversible rings. Moreover, we investigate some properties of these rings.

Erratum to: Generalized Semicommutative Rings

Erratum to: Generalized Semicommutative Rings

Generalized semicommutative rings

We call a ring R generalized semicommutative if, for any a, b ∈ R, ab = 0 only positive integers m and n exist such that amRbn = 0. It is shown that the class of generalized semicommutative rings lies in the class of central semicommutative rings and contains the class of weakly semicommutative-I rings, where the inclusions are strict. Relationships between generalized semicommutative rings and rings of other known types have been studied. We present a method for producing generalized semicommutative families from a given generalized semicommutative ring. We also provide several criteria for a generalized semicommutative ring to be reduced.

Generalized Semicommutative Rings

Generalized Semicommutative Rings

On a Generalized Semicommutative Ring

Let R be a ring with identity. In this paper, we introduce some new results in a class of rings which refers to generalization of semicommutative rings called Q-semicommutative rings whenever x2=0 implies xRx=0, for any a ∈R [7]. They study investigates general properties of Q-semicommutative rings and shows several results of semicommutative ring can be extended to Q-semicommutative rings

Strongly CM -semicommutative rings

We study the strongly semicommutative properties relative to a monoid crossed product. The concept of strongly $CM$-semicommutative rings is introduced and investigated. Many results related to semicommutative properties over polynomial rings, skew polynomial rings, monoid rings, and skew monoid rings are extended and unified.

On nil-McCoy rings relative to a monoid

The concept of nil-M-McCoy (nil-McCoy ring relative to monoid M), which are generalizations of McCoy ring and nil-M-Armendariz rings have been introduced, and we investigate their properties. It is shown that every NI ring is nil-M-McCoy for any unique product monoid M, it has also been shown that every semicommutative rings is nil-M-McCoy for any unique product monoid and any strictly totally ordered monoid M. Moreover, it is proved that for an ideal I of R, if I is semicommutative and R / I is nil-M-McCoy then R is nil-M-McCoy for any strictly totally ordered monoid. We extend and unify many known results related to McCoy rings and nil-Armendariz ring.

Special properties of the ring Sn(R)

Let [Formula: see text] be a ring with compatible endomorphisms [Formula: see text]. We investigate the quasi-Armendariz, Armendariz, McCoy, semicommutative properties and Property (A) of the ring [Formula: see text]. Our results yield more examples of quasi-Armendariz rings, Armendariz rings, McCoy rings, semicommutative rings and rings with Property (A).

On Some Generalizations of Reversible and Semicommutative Rings

The concept of strongly central reversible rings has been introduced in this paper. It has been shown that the class of strongly central reversible rings properly contains the class of strongly reversible rings and is properly contained in the class of central reversible rings. Various properties of the above-mentioned rings have been investigated. The concept of strongly central semicommutative rings has also been introduced and its relationships with other rings have been studied. Finally an open question raised in [D. W. Jung, N. K. Kim, Y. Lee and S. J. Ryu, Bull. Korean Math. Soc., 52(1) (2015), 247-261] has been answered.

Nil-Armendariz Condition on Skew Generalized Power Series Rings

Let R be a ring, S a strictly ordered monoid, and $$\omega :S \to {\text{End}}(R)$$ a monoid homomorphism. The skew generalized power series ring R[[S, ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Malcev–Neumann Laurent series rings. In the current work, we study the (S, ω)-nil Armendariz condition on R, a generalization of the standard nil-Armendariz condition from polynomials and power series to skew generalized power series. We resolve the structure of (S, ω)-nil Armendariz rings and obtain various necessary or sufficient conditions for a ring to be (S, ω)-nil Armendariz, unifying and generalizing a number of known nil Armendariz-like conditions in the aforementioned special cases. For example, we show that left uniserial nilpotent semicommutative rings are nil Armendariz. Moreover, we study on the relationship between the zip and weak zip properties of a ring R and these of the ring R[[S, ω]].

Reversibility of Rings with Respect to the Jacobson Radical

Let R be a ring with identity and J(R) denote the Jacobson radical of R. A ring R is called J-reversible if for any a, $$b \in R$$ , $$ab = 0$$ implies $$ba \in J(R)$$ . In this paper, we give some properties of J-reversible rings. We prove that some results of reversible rings can be extended to J-reversible rings for this general setting. We show that J-quasipolar rings, local rings, semicommutative rings, central reversible rings and weakly reversible rings are J-reversible. As an application it is shown that every J-clean ring is directly finite.

?-Armendariz Rings and Related Concepts

In this paper we investigated some new properties of ?-Armendariz rings and studied the relationships between ?-Armendariz rings and central Armendariz rings, nil-Armendariz rings, semicommutative rings, skew Armendariz rings, ?-compatible rings and others. We proved that if R is a central Armendariz, then R is ?-Armendariz ring. Also we explained how skew Armendariz rings can be ?-Armendariz, for that we proved that if R is a skew Armendariz?-compatible ring, then R is ?-Armendariz. Examples are given to illustrate the relations between concepts.

π-Armendariz Rings and Related Concepts

In this paper we investigated some new properties of π-Armendariz rings and studied the relationships between π-Armendariz rings and central Armendariz rings, nil-Armendariz rings, semicommutative rings, skew Armendariz rings, α-compatible rings and others. We proved that if R is a central Armendariz, then R is π-Armendariz ring. Also we explained how skew Armendariz rings can be ?-Armendariz, for that we proved that if R is a skew Armendariz π-compatible ring, then R is π-Armendariz. Examples are given to illustrate the relations between concepts.

Semicommutativity of the rings relative to prime radical

In this paper, we introduce a new kind of rings that behave like semicommutative rings, but satisfy yet more known results. This kind of rings is called $P$-semicommutative. We prove that a ring $R$ is $P$-semicommutative if and only if $R[x]$ is $P$-semicommutative if and only if $R[x, x^{-1}]$ is $P$-semicommutative. Also, if $R[[x]]$ is $P$-semicommutative, then $R$ is $P$-semicommutative. The converse holds provided that $P(R)$ is nilpotent and $R$ is power serieswise Armendariz. For each positive integer $n$, $R$ is $P$-semicommutative if and only if $T_n(R)$ is $P$-semicommutative. For a ring $R$ of bounded index $2$ and a central nilpotent element $s$, $R$ is $P$-semicommutative if and only if $K_s(R)$ is $P$-semicommutative. If $T$ is the ring of a Morita context $(A,B,M,N,\psi,\varphi)$ with zero pairings, then $T$ is $P$-semicommutative if and only if $A$ and $B$ are $P$-semicommutative. Many classes of such rings are constructed as well. We also show that the notions of clean rings and exchange rings coincide for $P$-semicommutative rings.