Mark L. Teply

Generalized deviation of posets and modular lattices

Abstract The existence and relationship of the Γ-deviation and dual Γ-deviation of a poset is investigated, where Γ is a nonempty set of linear order types. If Γ is a finite set of linear order types, then bounds are given for the Γ-deviation of a poset (when it exists). If Γ is an arbitrary set of indecomposable order types and L is a modular lattice, then the Γ-deviation behaves in a classical manner on subintervals of L. Examples are given to illustrate the results.

Prime ideals and radicals in semigroup-graded rings

This is a preprint version of a paper that will appear in Proc. Edinburgh Math. Soc. 39 (1996) 1–25. In this paper we study the ideal structure of the direct limit and direct sum (with a special multiplication) of a directed system of rings; this enables us to give descriptions of the prime ideals and radicals of semigroup rings and semigroup-graded rings. We show that the ideals in the direct limit correspond to certain families of ideals from the original rings, with prime ideals corresponding to “prime” families. We then assume the indexing set is a semigroup Ω with preorder defined by α ≺ β if β is in the ideal generated by α , and we use the direct sum to construct an Ω-graded ring; this construction generalizes the concept of a strong supplementary semilattice sum of rings. We show the prime ideals in this direct sum correspond to prime ideals in the direct limits taken over complements of prime ideals in Ω when two conditions are satisfied; one consequence is that when these conditions are satisfied, the prime ideals in the semigroup ring S[Ω] correspond bijectively to pairs (Φ, Q) with Φ a prime ideal of Ω and Q a prime ideal of S . The two conditions are satisfied in many bands and in any commutative semigroup in which the powers of every element become stationary. However, we show that the above correspondence fails when Ω is an infinite free band, by showing that S[Ω] is prime whenever S is. When Ω satisfies the above-mentioned conditions, or is an arbitrary band, we give a description of the radical of the direct sum of a system in terms of the radicals of the component rings for a class of radicals which includes the Jacobson radical and the upper nil radical. We do this by relating the semigroup-graded direct sum to a direct sum indexed by the largest semilattice quotient of Ω , and also to the direct product of the component rings.

The deviation, density, and depth of partially ordered sets

Let P be a poset, and let γ be a linear order type with |γ| ≥ 3. The γ-deviation of P, denoted by γ-dev P, is defined inductively as follows: (1) γ-dev P=0, if P contains no chain of order type γ; (2) γ-dev P = α, if γ-dev P ≮ α and each chain C of type γ in P contains elements a and b such that a<b and [a, b] as an interval of P has γ-deviation <α. There may be no ordinal α such that γ-dev P = α; i.e., γ-dev P does not exist. A chain is γ-dense if each of its intervals contains a chain of order type γ. If P contains a γ-dense chain, then γ-dev P fails to exist. If either (1) P is linearly ordered or (2) a chain of order type γ does not contain a dense interval, then the converse holds. For an ordinal ξ, a special set S(ξ) is used to study ωξ-deviation. The depth of P, denoted by δ(P) is the least ordinal β that does not embed in P*. Then the following statements are equivalent: (1) ωξ-dev P does not exist; (2) S(ξ) embeds in P; and (3) P has a subset Q of cardinality ℵξ such that δ(Q*) = ωξ + 1. Also ωξ-dev P = α<ωξ + 1 if and only if |δ(P*)|⩽ℵξ; if these equivalent conditions hold, then ωβξ < δ(P*) ≤ ωα + 1ξ for all β < α. Applications are made to the study of chains of submodules of a module over an associative ring.

Codivisible and Projective Covers

This paper continues the study of codivisible modules, whose definition is a “dualization” of Lambeks concept [4] of a divisible module relative to a torsion theory. The main purpose of this work is to give a solution to the following problem posed by Bland [2]: “It would be interesting to know under what conditions the universal existence of codivisible covers implies that of projective covers.” The if-and-only-if nature of the solution, which is given in Theorems 2 and 4, shows that our sufficient conditions are “best possible” conditions. The method of the solution introduces the concept of a pseudo-hereditary torsion theory, which may be of interest in its own right; in particular, every hereditary torsion theory and every faithful torsion theory is pseudo-hereditary. The main results for a pseudo-hereditary torsion theory (T, F) relate the conditions, R/T(R) is semiperfect and R/T(R) is left perfect, to the existence of codivisible covers (see Theorems 1 and 3 and Corollaries 1 and 2).

MODULAR QFD LATTICES WITH APPLICATIONS TO GROTHENDIECK CATEGORIES AND TORSION THEORIES

A modular lattice L with 0 and 1 is called quotient finite dimensional (QFD) if [x,1] has no infinite independent set for any x∈L. We extend some results about QFD modules to upper continuous modular lattices by using Lemonniers Lemma. One result says that QFD for a compactly generated lattice L is equivalent to Condition (C): for every m∈L, there exists a compact element t of L such that t∈[0,m] and [t,m[ has no maximal element. If L is not compactly generated, then QFD and (C) separate into two distinct conditions, which are analyzed and characterized for upper continuous modular lattices. We also extend to upper continuous modular lattices some characterizations of QFD modules with Gabriel dimension. Applications of these results are given to Grothendieck categories and module categories equipped with a torsion theory.

Torsion theoretic dimensions and relative Gabriel correspondence

Let τ be an hereditary torsion theory. For a ring with τ-Gabriel dimension, we find necessary and sufficient conditions for the existence of a bijective correspondence between the τ-torsionfree injective modules and the τ-closed prime ideals. As an application, new characterizations of fully bounded noetherian rings are obtained.

Right Hereditary, Right Perfect Rings are Semiprimary

If R is a right hereditary, right perfect ring, then every maximal ideal of R is idempotent, and every ideal of R has a stationary power. Consequently, every right hereditary, right perfect ring must be semiprimary.

Global dimensions of right coherent rings with left Krull dimension

The weak global dimension of a right coherent ring with left Krull dimension α ≥ 1 is found to be the supremum of the weak dimensions of the β-critical cyclic modules, where β I → assl gives a bijection between isomorphism classes on injective left R -modules and prime ideals of R , then the weak global dimension of R is the supremum of the weak dimensions of the simple left R -modules. These results are used to compute the left homological dimension of a right coherent, left noetherian ring. Some analogues of our results are also given for rings with Gabriel dimension.

The transfer of the Krull dimension and the Gabriel dimension to subidealizers

Let M be a right ideal of the ring T with identity. A unital subring R of T which contains M as a two-sided ideal is called a subidealizer ; the largest such subring is the idealizer I (M) of M in T. M is said to be generative if TM = T. In this case M is idempotent, and it follows from the dual basis lemma that T is finitely generated projective as a right i^-module (see [7, Lemma 2.1]) ; we make frequent use of these two facts in this paper. In recent years, techniques involving subidealizers and idealizers have been employed successfully in order to provide solutions to a number of ring theoretical questions. While most of the results involve only the right-hand side, left-handed properties have been studied as well (see [2] and [9]). In this paper we investigate the transfer of the Krull and Gabriel dimensions of T-modules when considered as i^-modules and of i^-modules when their tensor product with T is formed. For right modules this has been done in [6] for the case when M is semimaximal (that is, an intersection of a finite number of maximal right ideals). We obtain essentially the same results making use only of the hypotheses that M = M and that (T/M)R has finite length (for the Krull dimension) or (T/M)T and (R/M)R are semiartinian (for the Gabriel dimension). For left modules there are some results due to Teply [9] who studied the transfer of the Krull dimension for cyclic left modules. In view of the difficulties (see [2, p. 416]) concerning the transfer of left-handed chain conditions, it is surprising that the comparatively weak assumptions of R/M being left artinian (for the Krull dimension) and left semi-artinian (for the Gabriel dimension) assure an almost perfect transfer of these dimensions with one exception: passing down from T to R need not preserve the existence of the Krull dimension even if M is maximal, R/M is a field, and T is a left and right noetherian ring. In each of the sections, the results obtained for modules are applied to show that, with the exception mentioned above, the respective dimension of T equals that of R if either dimension exists. In this context we mention a recent theorem of Armendariz and Fisher ([1, Theorem 1.1]) which shows that in case M is generative the respective dimension of T is less than or equal to that of R, provided the latter exists. We briefly recall some of the definitions used in this article. The Krull dimension of a right 5-module M is defined recursively by setting K-dim (M) =

Generalized Deviations of Posets with Applications to Chain Conditions on Modules

The study of generalized deviations of a partially ordered set has its roots in the study of the Krull dimension of rings and modules. The concept of Krull dimension for commutative rings was developed by E. Noether and W. Krull in the 1920s. In 1923 E. Noether [27] explored the relationship between chains of prime ideals and dimensions of algebraic varieties. After five years, W. Krull [23] developed her idea into a powerful tool for arbitrary commutative rings satisfying the ascending chain condition for ideals. These rings are known today as Noetherian rings. Later, algebraists gave the name (classical) Krull dimension to the supremum of the length of finite chains of prime ideals in a ring.