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Structure of idempotents in rings without identity
Abstract
We study the structure of idempotents in polynomial rings, power series rings, concentrating in the case of rings without identity. In the procedure we introduce right Insertion-of-Idempotents-Property (simply, right IIP) and right Idempotent-Reversible (simply, right IR) as generalizations of Abelian rings. It is proved that these two ring properties pass to power series rings and polynomial rings. It is also shown that π-regular rings are strongly π-regular when they are right IIP or right IR. Next the noncommutative right IR rings, right IIP rings, and Abelian rings of minimal order are completely determined up to isomorphism. These results lead to methods to constructsuch kinds of noncommutative rings appropriate for the situations occurred naturally in studying standard ring theoretic properties.
... Nam Kyun Kim, Yang Lee, Yeonsook Seo in [KLS14] defined the notions of rings satisfying the right and left IIP properties as rings R in which rse = 0 implies res = 0 and ers = 0 implies res = 0 respectively, for any e ∈ E(R) and r, s ∈ R. These authors also defined the notions of rings satisfying the right and left IR properties as rings R in which re = 0 implies er = 0 and er = 0 implies re = 0 respectively, for any e ∈ E(R) and r ∈ R, and proved that an equivalent condition on a ring R to satisfy the right IR property is that ere = er holds for any e ∈ E(R) and r ∈ R. Note that in Theorem 2.7 we gave a deeper characterization of rings satisfying the right IR property. ...
... As we saw in Corollary 2.6, semicommutative rings satisfy the ICZ property. Nam Kyun Kim, Yang Lee, Yeonsook Seo in [KLS14,Examples 2.11] showed that semicommutative rings need not satisfy the IR property. ...
This note is intended as a discussion of some generalizations of the notion of a reversible ring, which may be obtained by the restriction of the zero commuta-tive property from the whole ring to some of its subsets. By the INCZ property we will mean the commutativity of idempotent elements of a ring with its nilpo-tent elements at zero, and by ICZ property we will mean the commutativity of idempotent elements of a ring at zero. We will prove that the INCZ property is equivalent to the abelianity even for nonunital rings. Thus the INCZ property implies the ICZ property. Under the assumption on the existence of unit, also the ICZ property implies the INCZ property. As we will see, in the case of nonunital rings, there are a few classes of rings separating the class of INCZ rings from the class of ICZ rings. We will prove that the classes of rings, that will be discussed in this note, are closed under extending to the rings of polynomials and formal power series.
... So |R| = 16. Every non-commutative abelian ring of minimal order is isomorphic to one of the rings in Example 2.14 by [13,Theorem 3.3]. But they are weakly reflexive by the argument in Example 2.14. ...
The concept of reflexive property is introduced by Mason. This note concerns a ring-theoretic property of matrix rings over reflexive rings. We introduce the concept of weakly reflexive rings as a generalization of reflexive rings. From any ring, we can construct weakly reflexive rings but not reflexive, using its lower nilradical. We study various useful properties of such rings in relation with ideals in matrix rings, showing that this new property is Morita invariant. We also investigate the weakly reflexive property of several sorts of ring extensions which have roles in ring theory. © 2018 Academy of Mathematics and Systems Science, Chinese Academy of Sciences.
In this article, we first observe a sort of property of ideals generated by zero-dividing polynomials over reversible rings, in relation with products of ideals at zero. As a natural consequence of this result, we introduce the concept of a partially reflexive ring that is related to the reflexive ring property. It is shown that abelian π-regular rings are partially reflexive when Jacobson radicals are nilpotent. A ring R, in which the Jacobson radical J(R) is nilpotent and R/J(R) is simple, is shown to be partially reflexive; and it is proved that the polynomial ring over R is also partially reflexive. The structures of several kinds of algebraic systems are investigated with respect to the partial reflexivity.
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.
The study of symmetric rings has important roles in ring theory and module theory. We investigate the structure of ring properties related to symmetric rings and introduce H-symmetric and π-symmetric as generalizations. We construct a non-symmetric reversible ring whose basic structure is infinite-dimensional, comparing with the finite-dimensional such rings of Anderson, Camillo and Marks. The structure of π-reversible rings (with or without identity) of minimal order is completely investigated. The properties of zero-dividing polynomials over IFP rings are studied more to show that polynomial rings over symmetric rings are π-symmetric. It is also proved that all conditions in relation with our arguments in this paper are equivalent for regular or locally finite rings.
Throughout this paper we consider associative rings with unity elements. In §1 various results on the representation of rings by rings of sections of special rings are compared. In particular, it is shown that results enunciated by Dauns and Hofmann, Koh, and the present author may all be deduced from one statement, the proof of which appears in §3.
Let R denote a near-ring such that for each x R, there exists an integer n(x) > 1 for which xn(x) = x. We show that the additive group of R is commutative if 0.x; = 0 for all x R and every non-trivial homomorphic image R¯ of R contains a non-zero idempotent e commuting multiplicatively with all elements of R¯. As the major consequence, we obtain the result that if R is distributively-generated, then R is a ring – a generalization of a recent theorem of Ligh on boolean near-rings.
Commutative rings form a very special subclass of rings, which shows quite different behaviour from the general case. For example, in a (non-trivial) commutative ring, the absence of zero-divisors is sufficient as well as necessary for the existence of a field of fractions, whereas for general rings, another infinite set of conditions is needed to characterize subrings of skew fields. This suggests the study of a class of rings which includes all commutative rings as well as all integral domains: reversible rings, where a ring is called reversible if ab = 0 implies ba = 0. It turns out that this condition helps to simplify other ring conditions, as we shall see in Section 2, although most of these results are at a somewhat superficial level. We therefore introduce a more technical notion, full reversibility, in Section 3, and show that this is the precise condition for the least matrix ideal to be proper and consist entirely of non-full matrices. Further, we show in Section 4 that a fully reversible ring is embeddable in a skew field if and only if it is an integral domain.
In what follows, all rings are associative, with a unit element 1 which is preserved by ring homomorphisms, inherited by subrings and acts unitally on modules.
I am grateful to V. de O. Ferreira for his comments, in particular the suggestion of using the notion of unit-stable rings. 1991 Mathematics Subject Classification 16U80.