On the applicability to semirings of two theorems from the theory of rings and modules

Problems concerning the extension of the Baer criterion for injectivity and embedding theorem of an arbitrary module over a ring into an injective module to the case of semirings are treated. It is proved that a semiring S satisfies the Baer criterion and every S-semimodule can be embedded in an injective semimodule if and only if S is a ring.     

Inequalities for Gondran-Minoux rank and idempotent semirings

The rank-sum, rank-product, and rank-union inequalities for Gondran-Minoux rank of matrices over idempotent semirings are considered. We prove these inequalities for matrices over quasi-selective semirings without zero divisors, which include matrices over the max-plus semiring. Moreover, it is shown that the inequalities provide the linear algebraic characterization for the class of quasi-selective semirings. Namely, it is proven that the inequalities hold for matrices over an idempotent semiring S without zero divisors if and only if S is quasi-selective. For any idempotent semiring which is not quasi-selective it is shown that the rank-sum, rank-product, and rank-union inequalities do not hold in general. Also, we provide an example of a selective semiring with zero divisors such that the rank-sum, rank-product, and rank-union inequalities do not hold in general.     

On Weak Dual Rickart Modules and Dual Baer Modules

We introduce and study the notion of wd-Rickart modules (i.e. modules M such that for every nonzero endomorphism ? of M, the image of ? contains a nonzero direct summand of M). We show that the class of rings R for which every right R-module is wd-Rickart is exactly that of right semi-artinian right V-rings. We prove that a module M is dual Baer if and only if M is wd-Rickart and M has the strong summand sum property. Several structure results for some classes of wd-Rickart modules and dual Baer modules are provided. Some relevant counterexamples are indicated.     

On quasi completely regular semirings

We extend the concepts of a completely π-regular semigroup and a GV semigroup to semirings and find a semiring analogue of a structure theorem on GV semigroups. We also show that a semiring S is quasi completely regular if and only if S is an idempotent semiring of quasi skew-rings.     

Large tilting modules and representation type

We study finiteness conditions on large tilting modules over arbitrary rings. We then turn to a hereditary artin algebra R and apply our results to the (infinite dimensional) tilting module L that generates all modules without preprojective direct summands. We show that the behaviour of L over its endomorphism ring determines the representation type of R. A similar result holds true for the (infinite dimensional) tilting module W that generates the divisible modules. Finally, we extend to the wild case some results on Baer modules and torsion-free modules proven in Angeleri Hügel, L., Herbera, D., Trlifaj, J.: Baer and Mittag-Leffler modules over tame hereditary algebras. Math. Z. 265, 1–19 (2010) for tame hereditary algebras.     

Weighted sums of squares in local rings and their completions, I

Let A be an excellent local ring of real dimension ≤2, let T be a finitely generated preordering in A, and let ${\widehat{T}}We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones. A right R-module M is called Baer if ${{\rm Ext}^{1}_{R}\,(M, T)\,=\,0}We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones. A right R-module M is called Baer if Ext¹_R (M, T) = 0{{\rm Ext}^{1}_{R}\,(M, T)\,=\,0} for all torsion modules T, and M is Mittag-Leffler in case the canonical map M?_R ?_{i ? I}Q_i? ?_{i ? I}(M?_RQ_i){M\otimes_R \prod _{i\in I}Q_i\to \prod _{i\in I}(M\otimes_RQ_i)} is injective where {Q_i}_{i ? I}{\{Q_i\}_{i\in I}} are arbitrary left R-modules. We show that a module M is Baer iff M is p-filtered where p is the preprojective component of the tame hereditary algebra R. We apply this to prove that the universal localization of a Baer module is projective in case we localize with respect to a complete tube. Using infinite dimensional tilting theory we then obtain a structure result showing that Baer modules are more complex then the (infinite dimensional) preprojective modules. In the final section, we give a complete classification of the Mittag-Leffler modules.     

On spectra of almost simple groups with symplectic or orthogonal socle

Finite groups are said to be isospectral if they have the same sets of the orders of elements. We investigate almost simple groups H with socle S, where S is a finite simple symplectic or orthogonal group over a field of odd characteristic. We prove that if H is isospectral to S, then H/S presents a 2-group. Also we give a criterion for isospectrality of H and S in the case when S is either symplectic or orthogonal of odd dimension.     

On semirings whose simple semimodules are projective

We study the semirings whose simple semimodules are all projective. In particular, we establish that for every semiring S this condition implies the injectivity of all simple S-semimodules and show that, in contrast to the case of rings, the projectivity of all simple semimodules in general is not a left-right symmetric property.     

Generalizations of von Neumann regular rings, <Emphasis Type="Italic">PP</Emphasis> rings,and Baer rings

We introduce the notions of IDS modules, IP modules, and Baer* modules, which are new generalizations of von Neumann regular rings, PP rings, and Baer rings, respectively, in a general module theoretic setting. We obtain some characterizations and properties of IDS modules, IP modules and Baer* modules. Some important classes of rings are characterized in terms of IDS modules, IP modules, and Baer* modules.     

Decompositions of modules lacking zero sums

A module over a semiring lacks zero sums (LZS) if it has the property that v +w = 0 implies v = 0 and w = 0. While modules over a ring never lack zero sums, this property always holds for modules over an idempotent semiring and related semirings, so arises for example in tropical mathematics.A direct sum decomposition theory is developed for direct summands (and complements) of LZS modules: The direct complement is unique, and the decomposition is unique up to refinement. Thus, every finitely generated “strongly projective” module is a finite direct sum of summands of R (assuming the mild assumption that 1 is a finite sum of orthogonal primitive idempotents of R). This leads to an analog of the socle of “upper bound” modules. Some of the results are presented more generally for weak complements and semi-complements. We conclude by examining the obstruction to the “upper bound” property in this context.     

FP-injective semirings,semigroup rings and Leavitt path algebras

In this paper we give characterisations of FP-injective semirings (previously termed “exact” semirings in work of the first author). We provide a basic connection between FP-injective semirings and P-injective semirings, and establish that FP-injectivity of semirings is a Morita invariant property. We show that the analogue of the Faith-Menal conjecture (relating FP-injectivity and self-injectivity for rings satisfying certain chain conditions) does not hold for semirings. We prove that the semigroup ring of a locally finite inverse monoid over an FP-injective ring is FP-injective and give a criterion for the Leavitt path algebra of a finite graph to be FP-injective.     

Direct sums of injective semimodules and direct products of projective semimodules over semirings

We prove that if the direct sum of a family of semimodules over a semiring S is an injective semimodule or if the direct product of a family of semimodules over S is a projective semimodule, then the cardinality of the subfamily consisting of all semimodules which are not modules is strictly less than the cardinality of S. As a consequence, we obtain semiring analogs of well-known characterizations of classical semisimple, quasi-Frobenius, and one-sided Noetherian rings.     

Modules having Baer summands

Let R be an arbitrary ring with identity and M a right R-module with S = End_R(M). Let F be a fully invariant submodule of M and I^?1(F) denotes the set {m∈M:Im?F} for any subset I of S. The module M is called F-Baer if I^?1(F) is a direct summand of M for every left ideal I of S. This work is devoted to the investigation of properties of F-Baer modules. We use F-Baer modules to decompose a module into two parts consists of a Baer module and a module determined by fully invariant submodule F, namely, for a module M, we show that M is F-Baer if and only if M = F⊕N where N is a Baer module. By using F-Baer modules, we obtain some new results for Baer rings.     

Hypersurface sections and S2-modules

We prove a hypersection section type criterion for a module over a noetherian local ring to satisfy Serre' condition S₂.     

Weakly regular modules over normal rings

Under study are some conditions for the weakly regular modules to be closed under direct sums and the rings over which all modules are weakly regular. For an arbitrary right R-module M, we prove that every module in the category σ(M) is weakly regular if and only if each module in σ(M) is either semisimple or contains a nonzero M-injective submodule. We describe the normal rings over which all modules are weakly regular.     

Periods of orbits modulo primes

Let S be a monoid of endomorphisms of a quasiprojective variety V defined over a global field K. We prove a lower bound for the size of the reduction modulo places of K of the orbit of any point α∈V(K) under the action of the endomorphisms from S. We also prove a similar result in the context of Drinfeld modules. Our results may be considered as dynamical variants of Artin's primitive root conjecture.     

Beurling type quotient modules over the bidisk and boundary representations

This paper mainly concerns Beurling type quotient modules of H²(D²) over the bidisk. By establishing a theorem of function theory over the bidisk, it is shown that a Beurling type quotient module is essentially normal if and only if the corresponding inner function is a rational inner function having degree at most (1,1). Furthermore, we apply this result to the study of boundary representations of Toeplitz algebras over quotient modules. It is proved that the identity representation of C^∗(Sz,Sw) is a boundary representation of B(Sz,Sw) in all nontrivial cases. This extends a result of Arveson to Toeplitz algebras on Beurling type quotient modules over the bidisk (cf. [W. Arveson, Subalgebras of C^∗-algebras, Acta Math. 123 (1969) 141-224; W. Arveson, Subalgebras of C^∗-algebras II, Acta Math. 128 (1972) 271-308]). The paper also establishes K-homology defined by Beurling type quotient modules over the bidisk.     

Covers of Semigroups and Function Semirings

We define and use a Galois correspondence to determine, for certain medial semigroups, S, some maximal semirings in the collection M(S) of self maps on S.