2023/06/06 by Engi̇n Büyükaşık, Engin Büyükaşık, Christian Lomp +1
Mathematics · #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Artinian ring #Combinatorics #Group (periodic table) #Group ring #Local ring #Mathematics #Noetherian #Noncommutative ring #Order (exchange) #Physics #Pure mathematics #Ring (chemistry) #Rings, Modules, and Algebras #Simple (philosophy) #Simple module #Von Neumann regular ring
paper · doi:10.1142/s0219498824502256
openalex publication_date 2023/06/06 · crossref created 2023/06/06 · openalex created_date 2023/06/07 · crossref issued 2023/07/10 · crossref published 2023/07/10 · crossref published-online 2023/07/10 · crossref published-print 2024/11/01 · crossref deposited 2024/11/05 · crossref indexed 2026/08/03 · openalex updated_date 2026/08/04
It is well known that a ring [Formula: see text] is right Kasch if each simple right [Formula: see text]-module embeds in a projective right [Formula: see text]-module. In this paper we study the dual notion and call a ring [Formula: see text] right dual Kasch if each simple right [Formula: see text]-module is a homomorphic image of an injective right [Formula: see text]-module. We prove that [Formula: see text] is right dual Kasch if and only if every finitely generated projective right [Formula: see text]-module is coclosed in its injective hull. Typical examples of dual Kasch rings are self-injective rings, V-rings and commutative perfect rings. Skew group rings of dual Kasch rings by finite groups are dual Kasch if the order of the group is invertible. Many examples are given to separate the notion of Kasch and dual Kasch rings. It is shown that commutative Kasch rings are dual Kasch, and a commutative ring with finite Goldie dimension is dual Kasch if and only if it is a classical ring (i.e. every element is a zero divisor or invertible). We obtain that, for a field [Formula: see text], a finite dimensional [Formula: see text]-algebra is right dual Kasch if and only if it is left Kasch. We also discuss the rings over which every simple right module is a homomorphic image of its injective hull, and these rings are termed strongly dual Kasch.