J-semicommutative环的性质(英文)

环R称为J-semicommutative若对任意a,b∈R由ab=0可以推得aRb∈J(R),这里J(R)是环R的Jacobson根.环R是J-semicommutative环当且仅当它的平凡扩张是J-semicommutative环当且仅当它的Dorroh扩张是J-semicommutative环当且仅当它的Nagata扩张是J-semicommutative环当且仅当它的幂级数环是J-semicommutative环.若R/J(R)是semicommutative环,则可得到R是J-semicommutative环.本文进一步论证了如果I是环R的一个幂零理想,且R/I是J-semicommutative环,则R也是J-semicommutative环.最后给出了J-semicommutative环与其他一些常见环的联系.

On Weakly Semicommutative Rings

A ring R is said to be weakly semicommutative if for any a,b∈R, ab=0 implies aRb（？）Nil（R）,where Nil（R） is the set of all nilpotent elements in R. In this note,we clarify the relationship between weakly semicommutative rings and NI-rings by proving that the notion of a weakly semicommutative ring is a proper generalization of NI-rings.We say that a ring R is weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical,and prove that if R is a weakly 2-primal ring which satisfiesα-condition for an endomorphismαof R（that is,ab=0（？）aα（b）=0 where a,b∈R） then the skew polynomial ring R[x;α] is a weakly 2-primal ring,and that if R is a ring and I is an ideal of R such that I and R/I are both weakly semicommutative then R is weakly semicommutative. Those extend the main results of Liang et al.2007（Taiwan Residents J.Math.,11（5）（2007）, 1359-1368） considerably.Moreover,several new results about weakly semicommutative rings and NI-rings are included.

Semicommutative Subrings of Matrix Rings

A ring R is called semicommutative if for every α∈ R, rR （α） is an ideal of R. It is well-known that the n by n upper triangular matrix ring is not semicommutative for any ring R with identity when n ≥ 2. We show that a special subring of upper triangular matrix ring over a reduced ring is semicommutative.

Weak M-Armendariz rings

For a monoid M, this paper introduces the weak M- Armendariz rings which are a common generalization of the M- Armendariz rings and the weak Armendariz rings, and investigates their properties. Moreover, this paper proves that： a ring R is weak M-Armendariz if and only if for any n, the n-by-n upper triangular matrix ring Tn （R） over R is weak M- Armendariz; if I is a semicommutative ideal of ring R such that R/I is weak M-Armendariz, then R is weak M-Armendariz, where M is a strictly totally ordered monoid; if a ring R is semicommutative and M-Armendariz, then R is weak M × N- Armendariz, where N is a strictly totally ordered monoid; a finitely generated Abelian group G is torsion-free if and only if there exists a ring R such that R is weak G-Armendariz.

Special Properties of Formal Triangular Matrix Rings

We consider the sufficient and necessary conditions for the formal triangular matrix ring being right minsymmetric, right DS, semicommutative, respectively.

Nilpotent Elements and Nil-Reflexive Property of Generalized Power Series Rings

Let R be a ring and (S,≤) a strictly ordered monoid. In this paper, we deal with a new approach to reflexive property for rings by using nilpotent elements, in this direction we introduce the notions of generalized power series reflexive and nil generalized power series reflexive, respectively. We obtain various necessary or sufficient conditions for a ring to be generalized power series reflexive and nil generalized power series reflexive. Examples are given to show that, nil generalized power series reflexive need not be generalized power series reflexive and vice versa, and nil generalized power series reflexive but not semicommutative are presented. We proved that, if R is a left APP-ring, then R is generalized power series reflexive, and R is nil generalized power series reflexive if and only if R/I is nil generalized power series reflexive. Moreover, we investigate ring extensions which have roles in ring theory.

Morita Context环的若干性质

研究由Morita Contexts所确定的环,讨论它的K-好环性质,弱Semicom-mutative性质,有许多单式正则元性质和广义稳定环性质等.

On Almost Armendariz Ring

In this paper,we introduce the notion of an almost Armendariz ring,which is a generalization of an Armendariz ring,and discuss some of its properties.It has been found that every almost Armendariz ring is weak Armendariz but the converse is not true.We prove that a ring R is almost Armendariz if and only if R[x]is almost Armendariz.It is also shown th at if R/I is an almost Armendariz ring and I is a semicommutative ideal,then H is an almost Armendariz ring.Moreover,the class of minimal non-commutative almost Armendariz rings is completely determined,up to isomorphism(minimal means having smallest cardinality).

The McCoy Condition on Skew Poincaré-Birkhoff-Witt Extensions

In this paper,we study the notion of McCoy ring over the class of non-commutative rings of polynomial type known as skew Poincare–Birkhoff–Witt extensions.As a consequence,we generalize several results about this notion considered in the literature for commutative rings and Ore extensions.

Dual Toeplitz Algebra on the Polydisk

In this paper,we prove the dual Toeplitz algebra I（C（D^-n））contains the ideal к of compact operators as its semicommutator ideal,and study its algebraic structure.We also get some results about spectrum of dual Toeplitz operators.