Semiperfect rings and Nakayama permutations

Abstract

Abstract. We study the conditions which force a semiperfect ring to admit a Nakayama permutation of its basic idempotents. We also give a few necessary and sufficient conditions for a semiperfect ring R, which cogenerates every 2-generated right R-module, to be right pseudo-Frobenius.2000 Mathematics Subject Classification. 16L30, 16L60.0. Introduction. Throughout R is an associative ring with identity and modules are unitary. The right and left annihilators of subset X of a ring R are denoted by r R ðXÞ and l R ðXÞ respectively. We write J ¼ JðRÞ for the Jacobson radical of a ring R and SocðMÞ for the socle of a module M. Right and left singular ideals of a ring R will be denoted by ZðR R Þ and Zð R RÞ respectively. By N j j j M we shall mean that N is an essential submodule of a module M.A ring R is called right mininjective (right principally injective) if every R-homomorphism from a minimal (principal) right ideal of R into R is given by left multiplication by an element of R. Mininjective rings were introduced by Harada [7] who studied them in Artinian case. Recently Nicholson and Yousif [13] studied arbitrary mininjective rings. Principally injective rings have been studied in [3,12,16,17]. A ring R is called right Kasch if R contains a copy of each simple right R-module. An idempotent e of a ring R is called local if eRe is a local ring; equivalently if eJ is the unique maximal submodule of eR. Nakayama [10] called a left and right Artinian ring R with basic set of idempotents e 1 ; :::; e n quasi-Frobenius if there exists a permutation of f1; :::; ng such that SocðRe ðiÞ Þ ffi Re i =Je i and Socðe i RÞ ffi e ðiÞ R=e ðiÞ J:Let R be a semiperfect ring with basic set of idempotents e 1 ; :::; e n . In this paper, following Nicholson and Yousif [12], we call a permutation of f1; :::; ng a Nakayama permutation if there exists a set k 1 ; :::; k n of elements of R such that for each i(1) Rk i Re ðiÞ and k i R e i R; (2) Rk i ffi Re i =Je i and k i R ffi e ðiÞ R=e ðiÞ J.In particular, fk 1 R; :::; k n Rg and fRk 1 ; :::; Rk n g are complete irredundant sets of representatives of isomorphism classes of simple right and simple left R-modules respectively and so R is left and right Kasch.If a ring is right self injective and right cogenerator, it is called right pseudoFrobenius (PF). Extending some well known results on PF and quasi-Frobenius (QF) rings, Nicholson and Yousif proved that a right minfull ring (that is a semiperfect right mininjective ring R with SocðeRÞ 6 ¼ 0 for every local idempotent e [13])

The Igusa–Todorov function for comodules

“…The function ν appears in contexts of categories of modules with finitely many simple modules (where it is bijective) and is usually called the Nakayama permutation (see for example [7]). Note that every cosemisimple coalgebra C verifies dim ϕ ( C M f ) = dim ϕ (M C f ) = 0, since every (right or left) comodule is injective.…”