NATURAL PARTIAL ORDERING ON E(HypG(2))

The concept of generalized hypersubstitutions was introduced by S. Leeratanavalee and K. Denecke as a way of making precise the concepts of strong hyperidentity and M-strong hyperidentity. The set Hyp G(2) of all generalized hypersubstitutions of type τ = (2) forms a monoid. All idempotent and regular elements in the monoid of all generalized hypersubstitutions of type τ = (2) were studied by W. Puninagool and S. Leeratanavalee. In this paper, we determine all primitive idempotent elements of this monoid and characterize the natural partial ordering on the set of all idempotent of this monoid.

HOW TO SHARPEN A TRIDIAGONAL PAIR

Let 𝔽 denote a field and let V denote a vector space over 𝔽 with finite positive dimension. We consider a pair of linear transformations A : V → V and A* : V → V that satisfy the following conditions: (i) each of A, A* is diagonalizable; (ii) there exists an ordering [Formula: see text] of the eigenspaces of A such that A* Vi ⊆ Vi-1 + Vi + Vi+1 for 0 ≤ i ≤ d, where V-1 = 0 and Vd+1 = 0; (iii) there exists an ordering [Formula: see text] of the eigenspaces of A* such that [Formula: see text] for 0 ≤ i ≤ δ, where [Formula: see text] and [Formula: see text]; (iv) there is no subspace W of V such that AW ⊆ W, A* W ⊆ W, W ≠ 0, W ≠ V. We call such a pair a tridiagonal pair on V. It is known that d = δ, and for 0 ≤ i ≤ d the dimensions of Vi, [Formula: see text], Vd-i, [Formula: see text] coincide. Denote this common dimension by ρi and call A, A*sharp whenever ρ0 = 1. Let T denote the 𝔽-subalgebra of End 𝔽(V) generated by A, A*. We show: (i) the center Z(T) is a field whose dimension over 𝔽 is ρ0; (ii) the field Z(T) is isomorphic to each of E0TE0, EdTEd, [Formula: see text], [Formula: see text], where Ei (resp. [Formula: see text]) is the primitive idempotent of A (resp. A*) associated with Vi (resp. [Formula: see text]); (iii) with respect to the Z(T)-vector space V the pair A, A* is a sharp tridiagonal pair.

The Cartan determinant and generalizations of quasihereditary rings

The Cartan determinant conjecture for left artinian rings was verified for quasihereditary rings showing detC(R) = detC(R/I), where I is a protective ideal generated by a primitive idempotent. This article identifies classes of rings generalizing the quasihereditary ones, first by relaxing the “projective” condition on heredity ideals. These rings, called left k-hereditary are all of finite global dimension. Next a class of rings is defined which includes left serial rings of finite global dimension, quasihereditary and left 1-hereditary rings, but also rings of infinite global dimension. For such rings, the Cartan determinant conjecture is true, as is its converse. This is shown by matrix reduction. Examples compare and contrast these rings with other known families and a recipe is given for constructing them.

Injective Modules Over Non-Artinian Serial Rings

A module is uniserial if its lattice of submodules is linearly ordered, and a ring R is left serial if R is a direct sum of uniserial left ideals. The following problem is considered. Suppose the injective hull of each simple left R-module is uniserial. When does this imply that the indecomposable injective left R-modules are uniserial? An affirmative answer is known when R is commutative and when R is Artinian. The following result is proved.Let R be a left serial ring and suppose that for each primitive idempotent e, eRe has indecomposable injective left modules uniserial. The following conditions are equivalent. (a) The injective hull of each simple left R-module is uniserial. (b) Every indecomposable injective left R-module is univerial. (c) Every finitely generated left R-module is serial.The rest of the paper is devoted to a study of some non-Artinian serial rings which serve to illustrate this theorem.

Nonsingular rings with essential socles

This paper is a study of nonsingular rings with essential socles. These rings were first investigated by Goldie [5] who studied the Artinian case and showed that an indecomposable nonsingular generalized uniserial ring is isomorphic to a full blocked triangular matrix ring over a sfield. The structure of nonsingular rings in which every ideal generated by a primitive idempotent is uniform was determined for the Artinian case by Gordon [6] and Colby and Rutter [2], and for the semiprimary case by Zaks [12]. Nonsingular rings with essential socles and finite identities were characterized by Gordon [7] and the author [10]. All these results were obtained by representing the rings in question as matrix rings. In this paper a matrix representation of arbitrary nonsingular rings with essential socles is found (section 2). The above results are special cases of this representation. A general method for representing rings as matrices is developed in section 1.

On central primitive idempotent measures

Let S be a compact semitopological semigroup and let P(S) be the convolution semigroup of probability measures on S. An idempotent measure μ in P(S) is defined to be primitive if and only idempotent measures in μP(S)μ are μ and the zero element m of P(S). In a previous paper [2] we give some characterization of primitive idempotent measures on S. Let Π(P(S)) be the set of primitive idempotents in P(S) and let Πc be the set of central primitive idempotents in P(S). It is shown in [1] that Π(P(S)) is neither an ideal nor even a subsemigroup of P(S) in general. The purpose of this paper is to investigate the structure of Πc.