LR-normal orthogroups

A new kind of regular orthogroups, namely, the LR-normal orthogroups is introduced and investigated. In particular, we will introduce the gearing techniques for fitting semigroup components on a semilattice in the construction of LR-normal orthogroups.     

A Structure Preserving Embedding of Semigroups Into Regular Semigroups

For any semigroup S a regular semigroup 𝒞(S) that embeds S can be constructed as the direct limit of a sequence of semigroups each of which contains a copy of its predecessor as a subsemigroup whose elements are regular. The construction is modified here to obtain an embedding of S into a regular semigroup R such that the nontrivial maximal subgroups of R are isomorphic to the Schützenberger groups of S and such that the restriction to S of any of Green's relations on R is the corresponding Green's relation on S.     

On the Embedding of Ordered Semigroups into Ordered Group

It was shown in [7] that any right reversible, cancellative ordered semigroup can be embedded into an ordered group and as a consequence, it was shown that a commutative ordered semigroup can be embedded into an ordered group if and only if it is cancellative. In this paper we introduce the concept of L-maher and R-maher semigroups and use a technique similar to that used in [7] to show that any left reversible cancellative ordered L or R-maher semigroup can be embedded into an ordered group.     

Ortho-u-monoids

In this paper, we study the class of ortho-u-monoids which are generalized orthogroups within the class of E(S)-semiabundant semigroups. After introducing the concept of (∼)-Green’s relations, and obtaining some important properties of (∼)-Green’s relations and super E(S)-semiabundant semigroups, we have given the semilattice decomposition of ortho-u-monoids and a structure theorem for regular ortho-u-monoids. The main techniques that we used in the study are the (∼)-Green’s relations, and the semi-spined product of semigroups.     

On maximal commutative subrings of non-commutative rings

We observe that every non-commutative unital ring has at least three maximal commutative subrings. In particular, non-commutative rings (resp., finite non-commutative rings) in which there are exactly three (resp., four) maximal commutative subrings are characterized. If R has acc or dcc on its commutative subrings containing the center, whose intersection with the nontrivial summands is trivial, then R is Dedekind-finite. It is observed that every Artinian commutative ring R, is a finite intersection of some Artinian commutative subrings of a non-commutative ring, in each of which, R is a maximal subring. The intersection of maximal ideals of all the maximal commutative subrings in a non-commutative local ring R, is a maximal ideal in the center of R. A ring R with no nontrivial idempotents, is either a division ring or a right ue-ring (i.e., a ring with a unique proper essential right ideal) if and only if every maximal commutative subring of R is either a field or a ue-ring whose socle is the contraction of that of R. It is proved that a maximal commutative subring of a duo ue-ring with finite uniform dimension is a finite direct product of rings, all of which are fields, except possibly one, which is a local ring whose unique maximal ideal is of square zero. Analogues of Jordan-Hölder Theorem (resp., of the existence of the Loewy chain for Artinian modules) is proved for rings with acc and dcc (resp., with dcc) on commutative subrings containing the center. A semiprime ring R has only finitely many maximal commutative subrings if and only if R has a maximal commutative subring of finite index. Infinite prime rings have infinitely many maximal commutative subrings.     

Gelfand Factor Rings and Weak Zariski Topologies

Let R be any ring with identity. Let N(R) (resp. J(R)) denote the prime radical (resp. Jacobson radical) of R, and let Spec _r (R) (resp. Spec _l (R), Max _r (R), Prim _r (R)) denote the set of all right prime ideals (resp. all left prime ideals, all maximal right ideals, all right primitive ideals) of R. In this article, we study the relationships among various ring-theoretic properties and topological conditions on Spec _r (R) (with weak Zariski topology). The following results are obtained: (1) R/N(R) is a Gelfand ring if and only if Spec _r (R) is a normal space if and only if Spec _l (R) is a normal space; (2) R/J(R) is a Gelfand ring if and only if every right prime ideal containing J(R) is contained in a unique maximal right ideal.     

Nonnoetherian coordinate rings with unique maximal depictions

A depiction of a nonnoetherian integral domain R is a special coordinate ring that provides a framework for describing the geometry of R. We show that if R is noetherian in codimension 1, then R has a unique maximal depiction T. In this case, the geometric dimensions of the points of Spec R may be computed directly from T. If in addition R has a normal depiction S, then S is the unique maximal depiction of R.     

LR-normal orthogroups

A new kind of regular orthogroups, namely, the LR-normal orthogroups is introduced and investigated. In particular, we will introduce the gearing techniques for fitting semigroup components on a semilattice in the construction of LR-normal orthogroups.     

Subgroups of GL 1(R) for Local Rings R

Abstract

Let R be a local ring, with maximal ideal m , and residue class division ring R/ m ?=?D. Denote by R*?=?G L ₁(R), the group of units of R. Here we investigate some algebraic structure of subnormal and maximal subgroups of R*. For instance, when D is of finite dimension over its center, it is shown that finitely generated subnormal subgroups of R* are central. It is also proved that maximal subgroups of R* are not finitely generated. Furthermore, assume that P is a nonabelian maximal subgroup of R* such that P contains a noncentral soluble normal subgroup of finite index, it is shown that D is a crossed product division algebra.     

Efficient Generation of Prime Ideals in Polynomial Rings up to Radical

We call an ideal I of a ring R radically perfect if among all ideals whose radical is equal to the radical of I, the one with the least number of generators has this number of generators equal to the height of I. Let R be a ring and R[X] be the polynomial ring over R. We prove that if R is a strong S-domain of finite Krull dimension and if each nonzero element of R is contained in finitely many maximal ideals of R, then each maximal ideal of R[X] of maximal height is the J _max-radical of an ideal generated by two elements. We also show that if R is a Prüfer domain of finite Krull dimension with coprimely packed set of maximal ideals, then for each maximal ideal M of R, the prime ideal MR[X] of R[X] is radically perfect if and only if R is of dimension one and each maximal ideal of R is the radical of a principal ideal. We then prove that the above conditions on the Prüfer domain R also imply that a power of each finitely generated maximal ideal of R is principal. This result naturally raises the question whether the same conditions on R imply that the Picard group of R is torsion, and we prove this to be so when either R is an almost Dedekind domain or a Prüfer domain with an extra condition imposed on it.     

New Braided Crossed Categories and Drinfel'd Quantum Double for Weak Hopf Group Coalgebras

A simple and nice structure theorem for orthogroups was given by Petrich in 1987. In this paper, we consider a generalized orthogroup, that is, a quasi-completely regular semigroup with a band of idempotents in which its set of regular elements, namely, RegS, forms an ideal of S. A method of construction of such semigroups is provided and as a result, the Petrich structure theorem of orthogroups becomes an immediate corollary of our theorem on generalized orthogroups. An example of such generalized orthogroup is also constructed. This example provides some useful information for the construction of various kinds of quasi-completely regular semigroups.     

Integrally Closed Domains with Only Finitely Many Star Operations

We prove that an integrally closed domain R admits only finitely many star operations if and only if R satisfies each of the following conditions: (1) R is a Prüfer domain with finite character, (2) all but finitely many maximal ideals of R are divisorial, (3) only finitely many maximal ideals of R contain a nonzero prime ideal that is contained in some other maximal ideal of R, and (4) if P ≠ (0) is the largest prime ideal contained in a (necessarily finite) collection of maximal ideals of R, then the prime spectrum of R/P is finite.     

Stability of <Emphasis Type="Italic">C</Emphasis>-regularized Semigroups

Let T = (T(t))t≥0 be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x ∈ Y, σu(A, Cx), the set of all points λ ∈ iR to which (λ - A)^-1 Cx can not be extended holomorphically, is at most countable and σr(A) N iR = Ф, then T is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous senfigroups.     

Von neumaivn regularity of sf-rings

A ring R is called left (right) SF-ring if all simple left (right) R-modules are flat. It is proved that R is Von Neumann regular if R is a right SF-ring whoe maximal essential right ideals are ideals. This gives the positive answer to a qestion proposed by R. Yue Chi MIng in 1985, and a counterexample is given to settle the follwoing question in the negative: If R is an ERT ring which is one-sided V-ring, is R a left and right V-ring? Some other conditions are given for a SF-ring to be regular.     

Irreducibility in the Total Ring of Quotients

Let R be a ring whose total ring of quotients Q is von Neumann regular. We investigate the structure of R when it admits an ideal that is irreducible as a submodule of the total ring of quotients. We characterize those rings which contain a maximal ideal that is irreducible in its total ring of quotients Q. An integral domain has a Q-irreducible ideal which is a maximal ideal if and only if R is a valuation domain. We show that when the total ring of quotients of R is von Neumann regular, then having a maximal ideal that is Q-irreducible is equivalently to certain valuation like properties, including the property that the regular ideals are linearly ordered.     

Global Theory of Lattice-Finite Noetherian Rings

We introduce and study lattice-finite Noetherian rings and show that they form a onedimensional analogue of representation-finite Artinian rings. We prove that every lattice-finite Noetherian ring R has Krull dimension ≼ 1, and that R modulo its Artinian radical is an order in a semi-simple ring. Our main result states that maximal overorders of R exist and have to be Asano orders, while they need not be fully bounded. This will be achieved by means of an idempotent ideal I(R), an invariant or R which is new even for classical orders R. This ideal satisfies I(R) = R whenever R is maximal. Presented by H. Tachikawa     

Regular semiartinian rings

We study the structure of rings over which every right module is an essential extension of a semisimple module by an injective one. A ring R is called a right max-ring if every nonzero right R-module has a maximal submodule. We describe normal regular semiartinian rings whose endomorphism ring of the minimal injective cogenerator is a max-ring.     

Umt-domains and domains with prüfer integral closure

An integral domain R is said to be a UMT-domain if uppers to zero in R[X) are maximal t-ideals. We show that R is a UMT-domain if and only if its localizations at maximal tdeals have Prüfer integral closure. We also prove that the UMT-property is preserved upon passage to polynomial rings. Finally, we characterize the UMT-property in certian pullback constructions; as an application, we show that a domain has Prüfer integral closure if and only if all its overrings are UMT-domains.     

The Structure of Ortho-LC-Monoids

The super-r-wide semigroups (in [6 Guo , Y. Q. , Shum , K. P. , Gong , C. M. ( 2011 ). On (*, ～)-Green's relations and ortho-lc-monoids . Communications in Algebra 39 : 5 – 31 . [Google Scholar]], called super-r-ample semigroups) [ortho-lc-monoids] within the class of rpp semigroups form a kind of generalized completely regular semigroups [orthogroups]. In this article, some structure theorems of ortho-lc-monoids are established and some special ortho-lc-monoids such as orthocrypto-lc-monoids, lc-Clifford semigroups, which we defined, are considered; the semilattice decomposition of super-r-wide semigroups is given. As direct corollaries of the results that we obtained, some new structure theorems for ortho-c-monoids and orthogroups, different from [10, 13], are given, and hence, the structure theorem for ortho-lc-monoids that we established is not the direct generalization of the results in [10 Petrich , M. ( 1987 ). A structure theorem for completely regular semigroups . Proc. Amer. Math. Soc. 99 : 617 – 622 .[Crossref], [Web of Science ®] , [Google Scholar], 13 Ren , X. M. , Shum , K. P. ( 2007 ). On superabundant semigroups whose set of idempotents form a subsemigroup . Algebra Colloquium 14 : 215 – 228 .[Crossref] , [Google Scholar]]; the structure of orthocryptogroups is reobtained; the well-known Clifford theorem is further generalized.