Parainjectivity, paraprojectivity and artinian principal ideal rings

Abstract

In this work, we introduce the concepts of parainjectivity and paraprojectivity. We give some basic properties about them and we obtain some characterizations of artinian principal ideal rings.

Annihilator ideals of indecomposable modules of finite-dimensional pointed Hopf algebras of rank one

“…Note that χ −l χ s = χ s χ −l = χ τ l (s) for 0 l n − 1. We have that the degree of χ s is equal to the degree of χ τ l (s) , namely, χ s (1) = χ τ l (s) (1). Hence we obtain…”

“…They proved that R[x] is a principal ideal ring if and only if R is a finite direct product of finite fields. Lately Alvarado-García et al [1] introduced the concepts of parainjectivity and paraprojectivity, and they obtain some characterizations of artinian principal ideal rings.…”

“…(0), (1), (e 0 ), (e 1 ), (e 2 ), (e 3 ), (e 0 + e 1 ), (e 0 + e 2 ), (e 0 + e 3 ), (e 1 + e 2 ), (e 1 + e 3 ), (e 2 + e 3 ), (e 0 + e 1 + e 2 ), (e 0 + e 1 + e 3 ), (e 0 + e 2 + e 3 ), (e 1 + e 2 + e 3 ), (z), (ze 0 ), (ze 1 ), (ze 2 ), (ze 3 ), (z(e 0 + e 1 )), (z(e 0 + e 2 )), (z(e 0 + e 3 )), (z(e 1 + e 2 )), (z(e 1 + e 3 )), (z(e 2 + e 3 )), (z(e 0 + e 1 + e 2 )), (z(e 0 + e 1 + e 3 )), (z(e 0 + e 2 + e 3 )), (z(e 1 + e 2 + e 3 )), (z + e 0 ), (ze 1 + e 0 ), (ze 2 + e 0 ), (z + e 1 ), (ze 0 + e 1 ), (ze 3 + e 1 ), (z + e 2 ), (ze 0 + e 2 ), (ze 3 + e 2 ), (z + e 3 ), (ze 1 + e 3 ), (ze 2 + e 3 ), (z + e 0 + e 3 ), (ze 1 + e 0 + e 3 ), (ze 2 + e 0 + e 3 ), (z + e 1 + e 2 ), (ze 0 + e 1 + e 2 ), (ze 3 + e 1 + e 2 ).…”