It was proved by the author earlier that every orthogonal extension of a reduced ring R is a subring of Q(R), the maximal two sided ring of quotients oiR and the orthogonal completion of R, if it exists, is unique upto an isomorphism. Here, in Theorem 2, we prove that the orthogonal completion of R, if it exists, is a ring of right quotients Q F (R) of R with respect to an idempotent filter F of dense right ideals of R. Introduction. Abian [2] showed that the canonical order relation ' < ' of Boolean rings can be defined for reduced rings R (a ring with no nonzero nilpotent element) by writing a < b if ab -a 2 and this order relation makes R into a partially ordered multiplicative semigroup. Reduced rings under this relation ' < ' were studied by Abian [1] and Chacron [5] to characterise the direct produce of integral domains, division rings and fields. Their studies involved the concepts of orthogonal completeness and orthogonal completion of reduced rings. These two concepts, on their own merit, were studied by Burgess, Raphael and Stephenson [3], [4], [11]. They proved that reduced rings which have enough idempotents (/-dense) or satisfy certain chain conditions have an orthogonal completion. In this paper we shall provide a necessary and sufficient condition for a reduced ring to have an orthogonal completion.
In this paper, it is proved that a reduced ring R has an orthogonal completion if and only if for every idempotent e e R, eR has an orthogonal completion. Every orthogonal subset X of R has a supremum in Q max(R), the maximal two sided ring of quotients of R, and the orthogonal completion of a reduced ring R, if it exists, is isomorphic to a unique subring of Q max(R). Hence the orthogonal completion of a reduced ring R, if it exists, is unique upto isomorphism. A reduced ring R has an orthogonal completion if and only if the collection of those elements of Q max(R) which are supremums of orthogonal subsets of R form a subring of Q max(R). Furthermore, every projectable ring R has an orthogonal completion , which is a Baer ring. It is also proved that for projectable rings R, where is the idempotent filter of those dense right ideals of R which contain a maximal orthogonal subset of idempotents of R.
In the present paper we are study of Matsumoto space on the projective algebra and Lie Algebra of the projective group. The projective Algebra of Matsumoto space is characterized as certain Lie sub algebra of the projective algebra. Further, which is devoted to studying the condition of Finsler space of constant flag curvature and vanishing S curvature admits a non Riemannian space of affine projective vector field with Matsumoto metric is Berwald space.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.