WEAK and CO-WEAK BAER MODULES: موديلات بيير الضعيفة والضعيفة المرافقة

The object of this paper is study the notions of weak Baer and weak Rickart rings and modules. We obtained many characterizations of weak Rickart rings and provide their properties. Relations ship between a weak Rickart (weak Baer) module and its endomorphism ring are studied. We proved that a weak Baer module with no infinite set of nonzero orthogonal idempotent elements in its endomorphism ring is precisely a Baer module. In addition, the endomorphism ring of a semi-projective weak Rickart module is semi-potent and the endomorphism ring of a semi-injective coweak Rickart module is semi-potent. Furthermore, we show that a free module is weak Baer if and only if its endomorphism ring is left weak Baer.

On Maximal Sets of Mutually Orthogonal Idempotent Latin Squares

It is a well-known trivial fact that for a given integer n there exists at most n — 2 pairwise orthogonal idempotent latin squares. In the following note we prove that for n a prime power there always exists n—2 such squares.

The analysis of randomized experiments with orthogonal block structure. II. Treatment structure and the general analysis of variance

Any orthogonal block structure for a set of experimental units has been previously shown to be expressible by a set of mutually orthogonal, idempotent matrices C i . When differential treatments are applied to the units, any linear model of treatment effects is expressible by another set of mutually orthogonal idempotent matrices T j . The analysis of any experiment having any set of treatments applied in any pattern whatever to units with an orthogonal block structure, is expressible in terms of the matrices C i , T j and the design matrix N, which lists the treatments applied to each unit. A unit-treatment additivity assumption and a valid randomization are essential to the validity of the analysis. The relevant estimation equations are developed for this general situation, and the idea of balance is given a generalized definition, which is illustrated by several examples. An outline is sketched for a general computer program to deal with the analysis of all experiments with an orthogonal block structure and linear treatment model.

The analysis of randomized experiments with orthogonal block structure. I. Block structure and the null analysis of variance

Nearly all the experimental designs so far proposed assume that the experimental units are or can be grouped together in blocks in ways that can be described in terms of nested and cross-classifications. When every nesting classification employed has equal numbers of subunits nested in each unit, then the experimental units are said to have a simple block structure . Every simple block structure has a complete randomization theory. Any permutation of the labelling of the units is permissible which preserves the block structure, and this defines a valid randomization procedure. In a null experiment all units receive the same treatment, and the population of all possible vectors of yields generated by the randomization procedure gives the null randomization distribution. The covariance matrix of this distribution, the null analysis of variance, and the expectations of the various mean squares in it are all derivable from the initial description of the block structure. All simple block structures can be characterized by a set of mutually orthogonal idempotent matrices C i . Such sets of matrices may also exist for non-simple block structures (e. g. those having unequal numbers of plots in a block), and the existence of such a set defines an orthogonal block structure. Non-simple orthogonal block structures do not have a complete randomization theory and inferences from them require further assumptions.