Narrow Results

For a finite lattice L, the congruence lattice Con L of L can be easily computed from the partially ordered set J(L) of join-irreducible elements of L and the join-dependency relation D_L on J(L). We establish a similar version of this result for the dimension monoid Dim L of L, a natural precursor of Con L. For L join-semidistributive, this result takes the following form:

Theorem 1. Let L be a finite join-semidistributive lattice. Then Dim L is isomorphic to the commutative monoid defined by generators Δ(p), for p ∈ J(L), and relations

As a consequence of this, we obtain the following results:

Theorem 2. Let L be a finite join-semidistributive lattice. Then L is a lower bounded homomorphic image of a free lattice iff Dim L is strongly separative, iff it satisfies the axiom

Theorem 3. Let A and B be finite join-semidistributive lattices. Then the box product A □ B of A and B is join-semidistributive, and the following isomorphism holds:

Let D be a principal ideal domain with quotient field F and suppose every residue field of D is finite. Let K be a finite separable field extension of F of degree at least 4 and let denote the integral closure of D in K. Let where f ∈ D is a nonzero nonunit. In this paper we show, assuming a mild condition on f, that cancellation of finitely generated modules fails for R, that is, there exist finitely generated R-modules L, M, and N such that L ⊕ M ≅ L ⊕ N and yet M ≇ N. In case the unit group of D is finite, we show that cancellation fails for almost all rings of the form , where p ∈ D is prime.

We consider those injective modules that determine every left exact preradical and we call them main injective modules. We construct a main injective module for every ring and we prove some of its properties. In particular we give a characterization, in terms of main injective modules, of rings with a dimension defined by a filtration in the lattice of left exact preradicals. We define also the concept of basic preradical and prove some of its properties. In particular we prove that the class of all basic preradicals is a set, giving a bijective correspondence with the set of all left exact preradicals.

We define the concept of "semicoprime" for preradicals and for submodules and we prove some properties that relate both of them. For any ring we define the ultrasocle preradical as a certain join of maximal semicoprime preradicals. It defines a kind of primary decomposition on modules. We compare the greatest semicoprime preradical, the meet of all unipotent preradicals, the socle preradical, and the ultrasocle preradical. We characterize rings which are finite product of simple rings in terms of some of these preradicals. We study the least semicoprime preradical above any preradical and we prove some of its properties. Using "Amitsur constructions" we define some related operators and prove some of their properties.

For bounded lattices, we introduce certain Galois connections, called (cyclically) essential, retractable and UC Galois connections, which behave well with respect to concepts of module-theoretic nature involving essentiality. We show that essential retractable Galois connections preserve uniform dimension, whereas essential retractable UC Galois connections induce a bijective correspondence between sets of closed elements. Our results are applied to suitable Galois connections between submodule lattices. Cyclically essential Galois connections unify semi-projective and semi-injective modules, while retractable Galois connections unify retractable and coretractable modules.

In this paper, we consider aspects of the big lattice of preradicals, related to pseudocomplements and supplements. We consider essential preradicals and superfluous preradicals, and we characterize the situation in which all nonzero preradicals are essential as well as the one in which all proper preradicals are superfluous. Also, we give two different proofs of the known fact that the big lattice of preradicals is strongly pseudocomplemented.

This paper is devoted to study the 2-absorbing preradicals and 2-absorbing submodules. We investigate some relations between the concept of 2-absorbing preradicals and 2-absorbing submodules using the lattice structure of preradicals.

In this paper, we define the concept of "co-2-absorbing" for preradicals and submodules. Also, we investigate the relationship between co-2-absorbing preradicals and co-2-absorbing submodules.

In this paper, some mappings to and from R-tors are introduced, and sufficient (or necessary) conditions for their being lattice isomorphisms are established.

A bounded poset $P:=(P,0,1,\le )$ is said to be lower finite if $P$ is infinite and for all $1\ne x\in P$, there are but finitely many $y\in P$ such that $y\le x.$ In this paper, we classify the modules $M$ over a commutative ring $R$ with identity with the property that the lattice $L_R(M)$ of $R$-submodules $M$ (under set-theoretic containment) is lower finite. Our results are summarized in Theorem 3.1 at the end of this note.

The theory of congruences on semigroups is an important part in the theory of semigroups. The aim of this paper is to study (2,1)-congruences on a glrac semigroup. It is proved that the (2,1)-congruences on a glrac semigroup become a complete sublattice of its lattice of congruences. Especially, the structures of left restriction semigroup (2,1)-congruences and the projection-separating (2,1)-congruences on a glrac semigroup are established. Also, we demonstrate that they are both complete sublattice of (2,1)-congruences and consider their relations with respect to complete lattice homomorphism.

Let $S(D)$ represent a set of proper nonzero ideals $I(D)$ (respectively, t-ideals $I_t(D)$) of an integral domain $D\ne qf(D)$ and let P be a valid property of ideals of D. We say S(D) meets P (denoted $S(D)◃P)$ if each $s\in S(D)$ is contained in an ideal satisfying P. If S(D) $◃P,$$dim(D)$ cannot be controlled. When $R=D[X],$$I(D)$$◃P$ does not imply I(R) $◃P$ while $I_t(D)$$◃P$ implies $I_t(R)$$◃P$ usually. We say S(D) meets P with a twist $($written $S(D){◃}^{t}P)$ if each $s\in S(D)$ is such that, for some $n\in N,$$s^n$ is contained in an ideal satisfying P and study $S(D){◃}^{t}P,$ as its predecessor. A modification of the above approach is used to give generalizations of almost bezout domains.

A differential operator of weight $\lambda$ is the algebraic abstraction of the difference quotient $d_{\lambda }(f)(x):=\left(\right.f(x+\lambda )-f(x)\left)\right./\lambda$, including both the derivation as $\lambda$ approaches to $0$ and the difference operator when $\lambda =1$. Correspondingly, differential algebra of weight $\lambda$ extends the well-established theories of differential algebra and difference algebra. In this paper, we initiate the study of differential operators with weights, in particular difference operators, on lattices. We show that differential operators of weight $-1$ on a lattice coincide with differential operators, while differential operators are special cases of difference operators. Distributivity of a lattice is characterized by the existence of certain difference operators. Furthermore, we characterize and enumerate difference operatorson finite chains and finite quasi-antichains.