On subgroups of residually nilpotent metabelian groups

ABSTRACT

We study conditions on an ideal A of a self-injective R such that the factor ring R/ A is again self-injective, extending certain of our results for PF rings (Faith, 2006 Faith , C. ( 2006 ). Factor rings of pseudo-Frobenius rings . J. Algebra and Its Applications 6 :(to appear). [CSA] [Web of Science ®] , [Google Scholar]). We also consider the same question for p -injective, and for CS -rings. For the CS -rings we consider conditions under which A splits off as a ring direct factor, equivalently, when A is generated by a central idempotent. Definitive results are obtained for an ideal A which is semiprime as a ring, that is, has no nilpotent ideals except zero, and which is a right annihilator ideal. Then A is said to be an r -semiprime right annulet ideal, and is generated by a central idempotent in the following cases: (1) whenever A is generated by an idempotent as a right (or left) ideal (Theorems 3.4, 3.6); (2) in any Baer ring R (Theorem 3.5); (3) in any right and left CS -ring R (Theorem 4.2), and (4) in any right nonsingular right CS -ring R (Theorem 5.5).

These results also generalize results of the author in Faith (1985 Faith , C. ( 1985 ). The maximal regular ideal of self-injective and continuous rings splits off . Arch. Math. 44 : 511 – 521 . [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]), where it is proven that the maximal regular ideal M( R) splits off in any right and left continuous ring.

The results are applied in Section 6 to extend theorems of Faith (1996 Faith , C. ( 1996 ). New characterizations of von Neumann regular rings and a conjecture of Shamsuddin . Publ. Mat. 40 : 383 – 385 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) characterizing VNR rings, and, as the title of Faith (1996 Faith , C. ( 1996 ). New characterizations of von Neumann regular rings and a conjecture of Shamsuddin . Publ. Mat. 40 : 383 – 385 . [CSA] [Crossref], [Web of Science ®] , [Google Scholar]) suggests, extend the conjecture of Shamsuddin.     

Loewy Modules with Finite Loewy Invariants and Max Modules with Finite Radical Invariants

Over a commutative ring R, a module is artinian if and only if it is a Loewy module with finite Loewy invariants [5 Facchini , A. ( 1981 ). Loewy and artinian modules over commutative rings . Ann. Mat. Pura Appl. 128 : 359 – 374 .[Crossref], [Web of Science ®] , [Google Scholar]]. In this paper, we show that this is not necesarily true for modules over noncommutative rings R, though every artinian module is always a Loewy module with finite Loewy invariants. We prove that every Loewy module with finite Loewy invariants has a semilocal endomorphism ring, thus generalizing a result proved by Camps and Dicks for artinian modules [3 Camps , R. , Dicks , W. ( 1993 ). On semilocal rings . Israel J. Math. 81 : 203 – 211 .[Crossref], [Web of Science ®] , [Google Scholar]]. Finally, we obtain similar results for the dual class of max modules.     

Laurent Series Ring over Semiperfect Ring Can Not be Semiperfect

One of the main results of the article [2 Sonin , K. I. ( 1996 ). Semiprime and semiperfect rings of Laurent series . Mathematical Notes 60 : 222 – 226 .[Crossref], [Web of Science ®] , [Google Scholar]] says that, if a ring R is semiperfect and ? is an authomorphism of R, then the skew Laurent series ring R((x, ?)) is semiperfect. We will show that the above statement is not true. More precisely, we will show that, if the Laurent series ring R((x)) is semilocal, then R is semiperfect with nil Jacobson radical.     

On Strongly Clean Matrix and Triangular Matrix Rings

A ring R with identity is called “clean” if every element of R is the sum of an idempotent and a unit, and R is called “strongly clean” if every element of R is the sum of an idempotent and a unit that commute. Strongly clean rings are “additive analogs” of strongly regular rings, where a ring R is strongly regular if every element of R is the product of an idempotent and a unit that commute. Strongly clean rings were introduced in Nicholson (1999 Nicholson , W. K. (1999). Strongly clean rings and Fitting's lemma. Comm. Algebra 27:3583–3592. [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) where their connection with strongly π-regular rings and hence to Fitting's Lemma were discussed. Local rings and strongly π-regular rings are all strongly clean. In this article, we identify new families of strongly clean rings through matrix rings and triangular matrix rings. For instance, it is proven that the 2 × 2 matrix ring over the ring of p-adic integers and the triangular matrix ring over a commutative semiperfect ring are all strongly clean.     

The Zero Divisor Graphs of Commutative Local Rings of Order p 4 and p 5

To each commutative ring R we can associate a zero divisor graph whose vertices are the zero divisors of R and such that two vertices are adjacent if their product is zero. Detecting isomorphisms among zero divisor graphs can be reduced to the problem of computing the classes of R under a suitable semigroup congruence. Presently, we introduce a strategy for computing this quotient for local rings using knowledge about a generating set for the maximal ideal. As an example, we then compute Γ(R) for several classes of rings; with the results in [4 Bloomfield , N. , Wickham , C. ( 2010 ). Local rings with genus 2 zero divisor graph . Comm. Alg. 38 ( 8 ): 2965 – 2980 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] these classes include all local rings of order p ⁴ and p ⁵ for prime p.     

A Few Properties of Just Infinite Algebras

In this article we prove a few interesting properties of just infinite algebras. Bartholdi (2006 Bartholdi , L. ( 2006 ). Branch rings, thinned rings, tree enveloping rings . Israel J. Math. 154 : 93 – 139 .[Crossref], [Web of Science ®] , [Google Scholar]), defines a particular class of just infinite algebras and demonstrates various properties of these examples. One such property, which is tedious to prove for his specific examples, is primality. We prove here that, in fact, all just infinite algebras are prime. We then consider two corollaries of this theorem; one suggests a weaker definition of just infinite for finitely generated algebras and the other examines the specific case of just infinite algebras which also satisfy a polynomial identity.     

Special Properties of Rings of Skew Generalized Power Series

Let R be a ring, S a strictly ordered monoid, and ω: S → End(R) a monoid homomorphism. In [30 Marks , G. , Mazurek , R. , Ziembowski , M. ( 2010 ). A unified approach to various generalizations of Armendariz rings . Bull. Aust. Math. Soc. 81 : 361 – 397 .[Crossref], [Web of Science ®] , [Google Scholar]], Marks, Mazurek, and Ziembowski study the (S, ω)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. Following [30 Marks , G. , Mazurek , R. , Ziembowski , M. ( 2010 ). A unified approach to various generalizations of Armendariz rings . Bull. Aust. Math. Soc. 81 : 361 – 397 .[Crossref], [Web of Science ®] , [Google Scholar]], we provide various classes of nonreduced (S, ω)-Armendariz rings, and determine radicals of the skew generalized power series ring R[[S ^≤, ω]], in terms of those of an (S, ω)-Armendariz ring R. We also obtain some characterizations for a skew generalized power series ring to be local, semilocal, clean, exchange, uniquely clean, 2-primal, or symmetric.     

Strongly J-Clean Matrices Over Local Rings

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. We investigate, in this article, a single strongly J-clean 2 × 2 matrix over a noncommutative local ring. The criteria on strong J-cleanness of 2 × 2 matrices in terms of a quadratic equation are given. These extend the corresponding results in [8 Li , B. ( 2009 ). Strongly clean matrix rings over noncommutative local rings . Bull. Korean Math. Soc. 46 : 71 – 78 .[Crossref], [Web of Science ®] , [Google Scholar], Theorems 2.7 and 3.2], [9 Li , Y. ( 2007 ). Strongly clean matrix rings over local rings . J. Algebra 312 : 397 – 404 .[Crossref], [Web of Science ®] , [Google Scholar], Theorem 2.6], and [11 Yang , X. , Zhou , Y. ( 2008 ). Strongly cleanness of the 2 × 2 matrix ring over a general local ring . J. Algebra 320 : 2280 – 2290 .[Crossref], [Web of Science ®] , [Google Scholar], Theorem 7].     

On weakly clean rings

A ring is called clean if every element is a sum of a unit and an idempotent, while a ring is said to be weakly clean if every element is either a sum or a difference of a unit and an idempotent. Commutative weakly clean rings were first discussed by Anderson and Camillo [2 Anderson, D. D., Camillo, V. P. (2002). Commutative rings whose elements are a sum of a unit and idempotent. Commun. Algebra 30(7):3327–3336.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] and were extensively investigated by Ahn and Anderson [1 Ahn, M.-S., Anderson, D. D. (2006). Weakly clean rings and almost clean rings. Rocky Mountain J. Math. 36:783–798.[Crossref], [Web of Science ®] , [Google Scholar]], motivated by the work on clean rings. In this paper, weakly clean rings are further discussed with an emphasis on their relations with clean rings. This work shows new interesting connections between weakly clean rings and clean rings.     

A Note on Dimension Modules

In [2 Camillo , V. P. , Zelmanowitz , J. M. ( 1980 ). Dimension modules . Pacific J. Math. 91 : 249 – 261 .[Crossref], [Web of Science ®] , [Google Scholar]] Camillo and Zelmanowitz stated that rings all whose modules are dimension modules are semisimple Artinian. It seem however that the proof in [2 Camillo , V. P. , Zelmanowitz , J. M. ( 1980 ). Dimension modules . Pacific J. Math. 91 : 249 – 261 .[Crossref], [Web of Science ®] , [Google Scholar]] contains a gap and applies to rings with finite Goldie dimension only. In this paper we show that the result indeed holds for all rings with a basis as well as for all commutative rings with Goldie dimension attained.     

On Serial Noetherian Rings

It is well known that every serial Noetherian ring satisfies the restricted minimum condition. In particular, following Warfield (1975 Warfield , R. B. ( 1975 ). Serial rings and finitely presented modules . J. Algebra 37 : 187 – 222 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), such a ring is a direct sum of an Artinian ring and hereditary prime rings. The aim of this note is to show that every serial ring having the restricted minimum condition is Noetherian.     

On Prime Spectrums of Commutative Rings

We show that π-regular rings and clean rings can be completely characterized by topological properties of their prime spectrums respectively. In addition, we give some applications of those result. Among others, we improve the main result of Samei (2004 Samei , K. ( 2004 ). Clean elements in commutative reduced rings . Comm. Algebra 32 ( 9 ): 3479 – 3486 . [CSA] [CROSSREF] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and give a new criterion for a clean ring that a commutative ring is clean if and only if idempotents lifts modulo every radical ideal.     

Regularity of Analytic Algebra of Prime Characteristic

Let A be an analytic algebra over a field k of characteristic p > 0. In this article, for an analytic k-algebra we introduce the concept of analytic pⁿ-basis which generalizes the pⁿ-basis defined in [1 Furuya , M. , Niitsuma , H. ( 2002 ). Regularity criterion of Noetherian local rings of prime characteristic . J. Algebra 247 : 219 – 230 .[Crossref], [Web of Science ®] , [Google Scholar]], and the concept of an pⁿ-admissible field for an algebraic function field, we give regularity criteria and absolute regularity criteria for an analytic algebra A/k in terms of the higher differential algebra and analytic pⁿ-basis. The results are partial extension of our previous results for affine algebras to the case of analytic algebras (cf. [1 Furuya , M. , Niitsuma , H. ( 2002 ). Regularity criterion of Noetherian local rings of prime characteristic . J. Algebra 247 : 219 – 230 .[Crossref], [Web of Science ®] , [Google Scholar], 3 Furuya , M. , Niitsuma , H. (2006). Regular local rings essentially of finite type over fields of prime characteristic. J. Algebra 306:703–711.[Crossref], [Web of Science ®] , [Google Scholar]]), and these are partial generalization of results of Orbanz in the analytic case (cf. [9 Orbanz , U. ( 1973 ). Höhere Derivationen und Regularität . J. Reine Angew. Mathematik 262/263 : 194 – 204 .[Crossref] , [Google Scholar]]).     

Supercharacter Theory Constructions Corresponding to Schur Ring Products

Diaconis and Isaacs have defined the supercharacter theories of a finite group to be certain approximations to the ordinary character theory of the group [7 Diaconis , P. , Isaacs , I. M. ( 2008 ). Supercharacters and superclasses for algebra groups . Trans. Amer. Math. Soc. 360 : 2359 – 2392 .[Crossref], [Web of Science ®] , [Google Scholar]]. We make explicit the connection between supercharacter theories and Schur rings, and we provide supercharacter theory constructions which correspond to Schur ring products of Leung and Man [12 Leung , K. H. , Man , S. H. ( 1996 ). On Schur rings over cyclic groups, II . J. Algebra 183 : 273 – 285 .[Crossref], [Web of Science ®] , [Google Scholar]], Hirasaka and Muzychuk [10 Hirasaka , M. , Muzychuk , M. ( 2001 ). An elementary abelian group of rank 4 is a CI-group . J. Combin. Theory Ser. A 94 : 339 – 362 .[Crossref], [Web of Science ®] , [Google Scholar]], and Tamaschke [20 Tamaschke , O. ( 1970 ). On Schur-rings which define a proper character theory on finite groups . Math. Z. 117 : 340 – 360 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Double Ore Extensions Versus Iterated Ore Extensions

Motivated by the construction of new examples of Artin–Schelter regular algebras of global dimension four, Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]] introduced an algebra extension A _P [y ₁, y ₂; σ, δ, τ] of A, which they called a double Ore extension. This construction seems to be similar to that of a two-step iterated Ore extension over A. The aim of this article is to describe those double Ore extensions which can be presented as iterated Ore extensions of the form A[y ₁; σ₁, δ₁][y ₂; σ₂, δ₂]. We also give partial answers to some questions posed in Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Relationship between Rings and Hyperrings by Using the Notion of Fundamental Relations

In the theory of hyperrings, fundamental relations make a connection between hyperrings and ordinary rings. The fundamental relation α on hyperrings is studied by Davvaz and Vougiouklis in Communications in Algebra [8 Davvaz , B. , Vougiouklis , T. ( 2007 ). Commutative rings obtained from hyperrings (H _v -rings) with α*-relations . Comm. Algebra 35 ( 11 ): 3307 – 3320 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. Now, in this article, we obtain more results about this relation. In particular, we determine necessary and sufficient conditions such that the relation α is transitive.     

On the almost Gorenstein property of determinantal rings

In this paper, we investigate the question of when the determinantal ring R over a field k is an almost Gorenstein local/graded ring in the sense of [14 Goto, S., Takahashi, R., Taniguchi, N. (2015). Almost Gorenstein rings - towards a theory of higher dimension. J. Pure Appl. Algebra 219:2666–2712.[Crossref], [Web of Science ®] , [Google Scholar]]. As a consequence of the main result, we see that if R is a non-Gorenstein almost Gorenstein local/graded ring, then the ring R has a minimal multiplicity.     

On Some Artinian QF-3 Rings

Carl Faith in 2003 introduced and investigated an interesting class of rings over which every cyclic right module has Σ-injective injective hull (abbr., right CSI-rings) [5 Faith , C. ( 2003 ). When cyclic modules have Σ-injective hulls . Comm. Algebra 13 : 4161 – 4173 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. Inspired by this we investigate rings over which every cyclic right R-module has a projective Σ-injective injective hull. We show that a ring R satisfies this condition if and only if R is right artinian, the injective hull of R _R is projective and every simple right R-module is embedded in R _R . We also characterize right artinian rings in terms of injective faithful right ideals and right CSI-rings.     

Automorphism Groups of Schur Rings

In 1993, Muzychuk [23 Muzychuk , Mikhail E. ( 1993 ). The structure of rational Schur rings over cyclic groups . European Journal of Combinatorics 14 : 479 – 490 .[Crossref], [Web of Science ®] , [Google Scholar]] showed that the rational Schur rings over a cyclic group Z _n are in one-to-one correspondence with sublattices of the divisor lattice of n, or equivalently, with sublattices of the lattice of subgroups of Z _n . This can easily be extended to show that for any finite group G, sublattices of the lattice of characteristic subgroups of G give rise to rational Schur rings over G in a natural way. Our main result is that any finite group may be represented as the (algebraic) automorphism group of such a rational Schur ring over an abelian p-group, for any odd prime p. In contrast, over a cyclic group the automorphism group of any Schur ring is abelian. We also prove a converse to the well-known result of Muzychuk [24 Muzychuk , Mikhail E. ( 1994 ). On the structure of basic sets of Schur rings over cyclic groups . Journal of Algebra 169 : 655 – 678 .[Crossref], [Web of Science ®] , [Google Scholar]] that two Schur rings over a cyclic group are isomorphic if and only if they coincide; namely, we show that over a group which is not cyclic, there always exist distinct isomorphic Schur rings.