Abelian Ring Research Articles

Stable range conditions for abelian and duo rings

The article deals with the following question: when does the classical ring of quotientsof a duo ring exist and idempotents in the classical ring of quotients $Q_{Cl} (R)$ are thereidempotents in $R$? In the article we introduce the concepts of a ring of (von Neumann) regularrange 1, a ring of semihereditary range 1, a ring of regular range 1. We find relationshipsbetween the introduced classes of rings and known ones for abelian and duo rings.We proved that semihereditary local duo ring is a ring of semihereditary range 1. Also it was proved that a regular local Bezout duo ring is a ring of stable range 2. In particular, the following Theorem 1 is proved: For an abelian ring $R$ the following conditions are equivalent:$1.$\ $R$ is a ring of stable range 1; $2.$\ $R$ is a ring of von Neumann regular range 1.
 The paper also introduces the concept of the Gelfand element and a ring of the Gelfand range 1 for the case of a duo ring. Weproved that the Hermite duo ring of the Gelfand range 1 is an elementary divisor ring (Theorem 3).

Lifting potent elements modulo nil ideals

It is classically known that idempotents lift modulo nil one-sided ideals. So it is natural to ask if the same is true for potent elements. Although we answer this question in negative, we prove that the answer is positive in several special cases. For instance, the answer is positive in rings with finite characteristic. Let I be a nil one sided ideal of R and x ∈ R is such that x n + 1 − x ∈ I for some positive integer n . We prove that if n is a unit in R , then there exists an element p ∈ R with p n + 1 = p such that x − p ∈ I . By taking n = 1 , the result that idempotents lift modulo nil one sided ideals is retrieved. For a nil ideal I of an abelian ring R , we prove that potent elements lift modulo I precisely when torsion units or periodic elements lift modulo I . It follows that torsion units or periodic elements may also not lift modulo nil ideals. We prove that torsion units, potent elements and periodic elements lift modulo every nil ideal of a π -regular ring.

Structure of NI rings related to centers

We first obtain that NI rings satisfy a property that if ab is central for elements a, b, then for some , by applying a property of reduced rings. We prove next the following: Let R be a ring and I be the ideal of R generated by the subset . (i) Suppose that ab is central for and ab – ba is a nonzero nilpotent. Then, is a nonzero nilpotent ideal of the subring A of R, where 1 is the identity of R, , and A is the algebra generated by a, b over B. (ii) If R is NI, then I is nil and R/I is an Abelian NI ring. (iii) Let R be reversible and ab be central for . Then, there exists such that, for every and for all ; especially . We call a ring pseudo-NI if it satisfies the first property of NI rings to be mentioned and examine the structures of NI and pseudo-NI rings in several ring theoretic situations, showing that semisimple Artinian rings are pseudo-NI.

On Strongly $$J_n$$ J n -Clean Rings

A ring R is said to be strongly $$J_n$$ -clean if n is the least positive integer such that every element a is strongly $$J_n$$ -clean, that is, there exists an idempotent e such that $$ea=ae$$ , $$a-e$$ is a unit and $$ea^n$$ is in the Jacobson radical J(R). It is proved that strongly $$J_n$$ -clean ring is a strongly clean ring with stable range one. If R is an abelian ring (a ring in which all idempotents are central), then R is strongly $$J_n$$ -clean if and only if R / J(R) is strongly $$\pi $$ -regular and idempotents lift modulo J(R). Some examples and basic properties of these rings are studied. Some criterions in terms of solvability of the characteristic equation are obtained for such a $$2\times 2$$ matrix to be strongly $$J_2$$ -clean.

On a generalization of NC-McCoy rings

In the present paper we concentrate on a natural generalization of NC-McCoy rings that is called J-McCoy and investigate their properties. We prove that local rings are J-McCoy. Also, for an abelian ring R, we show that R is J-McCoy if and only if eR is J-McCoy, where e is an idempotent element of R. Moreover, we give an example to show that the J-McCoy property does not pass Mn(R), but S(R; n);A(R; n);B(R; n) and T(R; n) are J-McCoy

Structure of Abelian rings

Let R be a ring with identity. We use J(R); G(R); and X(R) to denote the Jacobson radical, the group of all units, and the set of all nonzero nonunits in R; respectively. A ring is said to be Abelian if every idempotent is central. It is shown, for an Abelian ring R and an idempotent-lifting ideal N ⊆ J(R) of R; that R has a complete set of primitive idempotents if and only if R/N has a complete set of primitive idempotents. The structure of an Abelian ring R is completely determined in relation with the local property when X(R) is a union of 2; 3; 4; and 5 orbits under the left regular action on X(R) by G(R): For a semiperfect ring R which is not local, it is shown that if G(R) is a cyclic group with 2 ∈ G(R); then R is finite. We lastly consider two sorts of conditions for G(R) to be an Abelian group.

Rings in which every element is either a sum or a difference of a nilpotent and an idempotent

Generalizing the notion of nil-cleanness from [A. J. Diesl, Nil clean rings, J. Algebra 383 (2013) 197–211], in parallel to [P. V. Danchev and W. Wm. McGovern, Commutative weakly nil clean unital rings, J. Algebra 425 (2015) 410–422], we define the concept of weak nil-cleanness for an arbitrary ring. Its comprehensive study in different ways is provided as well. A decomposition theorem of a weakly nil-clean ring is obtained. It is completely characterized when an abelian ring is weakly nil-clean. It is also completely determined when a matrix ring over a division ring is weakly nil-clean.

A non-Abelian black ring

We construct a supersymmetric black ring solution of SU(2) N=1, d=5 Super-Einstein–Yang–Mills (SEYM) theory by adding a distorted BPST instanton to an Abelian black ring solution of the same theory. The change cannot be observed from spatial infinity: neither the mass, nor the angular momenta or the values of the scalars at infinity differ from those of the Abelian ring. The entropy is, however, sensitive to the presence of the non-Abelian instanton, and it is smaller than that of the Abelian ring, in analogy to what happens in the supersymmetric colored black holes recently constructed in the same theory and in N=2, d=4 SEYM. By taking the limit in which the two angular momenta become equal we derive a non-Abelian generalization of the BMPV rotating black-hole solution.

INSERTION-OF-FACTORS-PROPERTY WITH FACTORS NILPOTENTS

We in this note study a ring theoretic property which unies Armendariz and IFP. We call this new concept INFP. We rst show that idempotents and nilpotents are connected by the Abelian ring property. Next the structure of INFP rings is studied in relation to several sorts of algebraic systems.

On the Structures of Abelianπ-Regular Rings

Assume thatRis an Abelian ring. In this paper, we characterize the structure ofRwheneverRisπ-regular. It is also proved that an Abelianπ-regular ring is isomorphic to the subdirect sum of some metadivision rings.

USING IDEALS TO PROVIDE A UNIFIED APPROACH TO UNIQUELY CLEAN RINGS

AbstractIn this article, we introduce the notion of the uniquely $I$-clean ring and show that, if $R$ is a ring and $I$ is an ideal of $R$ then $R$ is uniquely $I$-clean if and only if ($R/ I$ is Boolean and idempotents lift uniquely modulo $I$) if and only if (for each $a\in R$ there exists a central idempotent $e\in R$ such that $e- a\in I$ and $I$ is idempotent-free). We examine when ideal extension is uniquely clean relative to an ideal. Also we obtain conditions on a ring $R$ and an ideal $I$ of $R$ under which uniquely $I$-clean rings coincide with uniquely clean rings. Further we prove that a ring $R$ is uniquely nil-clean if and only if ($N(R)$ is an ideal of $R$ and $R$ is uniquely $N(R)$-clean) if and only if $R$ is both uniquely clean and nil-clean if and only if ($R$ is an abelian exchange ring with $J(R)$ nil and every quasiregular element is uniquely clean). We also show that $R$ is a uniquely clean ring such that every prime ideal of $R$ is maximal if and only if $R$ is uniquely nil-clean ring and $N(R)= {\mathrm{Nil} }_{\ast } (R)$.

Inner inverses and inner annihilators in rings

For any ring element α∈R, we study the group of inner annihilators IAnn(α)={p∈R:αpα=0} and the set I(α) of inner inverses of α. For any Jacobson pair α=1−ab and β=1−ba, the groups A=IAnn(α) and B=IAnn(β) are shown to be equipotent, and A⊕C is shown to be group isomorphic to B⊕C where C=Annℓ(α)⊕Annr(α). In the case where α is (von Neumann) regular, we show further that A≅B as groups. For any Jacobson pair {α,β}, a “new Jacobson map” Φ:I(α)→I(β) is constructed that is a semigroup homomorphism with respect to the von Neumann product, and preserves units, reflexive inverses and commuting inner inverses. In particular, for any abelian ring R, Φ is a semigroup isomorphism between I(α) and I(β). As a byproduct of our methods, we also show that a ring R satisfies internal cancellation iff every Jacobson pair of regular elements are equivalent over R. In particular, the latter property holds for many rings, including semilocal rings, unit-regular rings, strongly π-regular rings, and finite von Neumann algebras.

Classes of Almost Clean Rings

A ring is clean (almost clean) if each of its elements is the sum of a unit (regular element) and an idempotent. A module is clean (almost clean) if its endomorphism ring is clean (almost clean). We show that every quasi-continuous and nonsingular module is almost clean and that every right CS (i.e. right extending) and right nonsingular ring is almost clean. As a corollary, all right strongly semihereditary rings, including finite AW *-algebras and noetherian Leavitt path algebras in particular, are almost clean. We say that a ring R is special clean (special almost clean) if each element a can be decomposed as the sum of a unit (regular element) u and an idempotent e with aR ∩ eR = 0. The Camillo-Khurana Theorem characterizes unit-regular rings as special clean rings. We prove an analogous theorem for abelian Rickart rings: an abelian ring is Rickart if and only if it is special almost clean. As a corollary, we show that a right quasi-continuous and right nonsingular ring is left and right Rickart. If a special (almost) clean decomposition is unique, we say that the ring is uniquely special (almost) clean. We show that (1) an abelian ring is unit-regular (equiv. special clean) if and only if it is uniquely special clean, and that (2) an abelian and right quasi-continuous ring is Rickart (equiv. special almost clean) if and only if it is uniquely special almost clean. Finally, we adapt some of our results to rings with involution: a *-ring is *-clean (almost *-clean) if each of its elements is the sum of a unit (regular element) and a projection (self-adjoint idempotent). A special (almost) *-clean ring is similarly defined by replacing “idempotent” with “projection” in the appropriate definition. We show that an abelian *-ring is a Rickart *-ring if and only if it is special almost *-clean, and that an abelian *-ring is *-regular if and only if it is special *-clean.

Quasi-Normal Rings

A ring R is defined to be quasi-normal if ae = 0 implies eaRe = 0 for a ∈ N(R) and e ∈ E(R), where E(R) and N(R) stand, respectively, for the set of idempotents and the set of nilpotents of R. It is proved that R is quasi-normal if and only if eR(1 − e)Re = 0 for each e ∈ E(R) if and only if T n (R, R) is quasi-normal for any positive integer n. And it follows that for a quasi-normal ring R, R is Π-regular if and only if N(R) is an ideal of R and R/N(R) is regular. Also, using quasi-normal ring, we proved the following: (1) R is an abelian ring if and only if R is a quasi-normal left idempotent reflexive ring; (2) R is a strongly regular ring if and only if R is a von Neumann regular quasi-normal ring; (3) Let R be a quasi-normal ring. Then R is a clean ring if and only if R is an exchange ring; (4) Let R be a quasi-normal Π-regular ring. Then R is a (S,2)-ring if and only if ℤ/2ℤ is not a homomorphic image of R.

Clean elements in abelian rings

Let R be a ring with identity. An element in R is said to be clean if it is the sum of a unit and an idempotent. R is said to be clean if all of its elements are clean. If every idempotent in R is central, then R is said to be abelian. In this paper we obtain some conditions equivalent to being clean in an abelian ring.

Spectrum of a Noncommutative Ring

R is any ring with identity. Let Spec r (R) (resp. Spec(R)) be the set of all prime right ideals (resp. all prime ideals) of R and let U r (eR) = {P ∈ Spec r (R) | e ∉ P}. In this article, we study the relationships among various ring-theoretic properties and topological conditions on Spec r (R) (with weak Zariski topology). A ring R is called Abelian if all idempotents in R are central (see Goodearl, 1991). A ring R is called 2-primal if every nilpotent element is in the prime radical of R (see Lam, 2001). It will be shown that for an Abelian ring R there is a bijection between the set of all idempotents in R and the clopen (i.e., closed and open) sets in Spec r (R). And the following results are obtained for any ring R: (1) For any clopen set U in Spec r (R), there is an idempotent e in R such that U = U r (eR). (2) If R is an Abelian ring or a 2-primal ring, then, for any idempotent e in R, U r (eR) is a clopen set in Spec r (R). (3) Spec r (R) is connected if and only if Spec(R) is connected.

ABELIAN EXCHANGE MODULES#

ABSTRACT Let Mk be a right k-module with endomorphism ring E=End(Mk). We prove that if E is an Abelian exchange ring, then Mk has the full exchange property. We also give an extension of this result in the case E is regular.

On the invertible objects in tensor categories

The invertible objects in a tensor category form a subcategory the Grothendieck ring of which is the group ring of an abelian group. This abelian fusion ring acts on the objects of the initial category and one can in principle determine all 6j-symbols that contain the lable of an invertible object.

Algebras with three anticommuting elements. II. Two algebras over a singular field

The Clifford algebra Ω generated by the elements {1,ω} with (ω)2 = +1, is an abelian ring of dimension two with properties analogous to the complex field C. The ring Ω has a string of singular inverses, and may be regarded as a ’’singular field’’ which circumvents both the fundamental theorem of algebra and the Frobenius theorem. We construct two associative algebras of dimension four over Ω: the Clifford algebra Ω1 and the biquaternions of Clifford Ω2, and demonstrate that both algebras possess inverses everywhere except on a singular region akin to the light cone of the Minkowski space. Matrix representations are discussed, as well as the importance of the algebras Ω1 and Ω2, in the description of physical vector fields.