Let R and S be arbitrary associative rings. Given a bimodule RWS , we denote by ∆? and Γ? the functors Hom?(−, W ) and Ext?(−, W ), where ? = R or S. We say that RWS is a finitistic weakly cotilting bimodule (briefly FWC) if for each module M cogenerated by W , finitely generated or homomorphic image of a finite direct sum of copies of W , ΓM = 0 = Ext(M, W ). We are able to describe, on a larg...

Exponent matrices appeared in the study of tiled orders over discrete valuation rings. Many properties of such orders are formulated using this notion. We think that such matrices are of interest in them own right, in particular, it is convenient to write finite partially ordered sets (posets) and finite metric spaces as special exponent matrices. Note that when we defined a quiver Q(E) of a re...

We give necessary and sufficient conditions for a skew polynomial ring K[t; σ, δ] over a division ring K to be a left V -domain. In particular, when this ring admits a unique simple left module, the conditions obtained include: 1) all polynomials are Wedderburn, 2) all n× n matrices over K are (σ, δ)-similar. We also provide necessary and sufficient conditions for this ring to be both left and ...

The object of this paper is to study the relationship between certain projective modules and their endomorphism rings. Specifically, the basic problem is to describe the projective modules whose endomorphism rings are (von Neumann) regular, local semiperfect, or left perfect. Call a projective module regular if every cyclic submodule is a direct summand. Thus a ring is a regular module if it is...

We discuss representations of the projective line over a ring R with 1 in a projective space over some (not necessarily commutative) field K. Such a representation is based upon a (K,R)-bimodule U . The points of the projective line over R are represented by certain subspaces of the projective space P(K,U ×U) that are isomorphic to one of their complements. In particular, distant points go over...

Trinion and Quaternion numbers are one of the hypercomplex which is an extensions complex number. From numbers, a bimodule can be formed ordered pair Quaternion. Furthermore, number, their into Formal Triangle Matrix. The Matrix better known as Upper Since number rings, then called Triangular Ring. purpose this study to construct Armendariz Ring using obtained results will show that Rings -Skew...