The Almost Split Sequences for Trivial Extensions of Hereditary Algebras

Let A be a basic hereditary artin algebra and R = A Q be the trivial extension of A by its minimal injective cogenerator Q. We construct some right （left） almost split morphisms and irreducible morphisms in modR through the corresponding morphisms in modA. Furthermore, we can determine its almost split sequences in modR.

The Category of Modules on an n-Trivial Extension: the Basic Properties

In this paper we investigate a categorical aspect of n-trivial extension of a ring by a family of modules.Namely,we introduce the right(resp.,left)n-trivial extension of a category by a family of endofunctors.Among other results,projective,injective and flat objects of this category are characterized,and two applications are presented at the end of this paper.We characterize when an n-trivial extension ring is k-perfect and establish a result on the self-injective dimension of an n-trivial extension ring.

Generating Relations for Grothendieck Groups of Trivial Extension Algebras

In this paper we describe explicitly the generating relations for the Grothendieck groups of trivial extension algebras of tame hereditary algebras.

Trivial extension of Koszul algebras

Let A be a Koszul algebra, and let M be a graded A-bimodule. We prove that the trivial extension algebra of A by M is also a Koszul algebra whenever M is Koszul as a left A-module. Applications and examples are also provided.

Extensions of strongly π-regular general rings

The concept of the strongly π-regular general ring （with or without unity） is introduced and some extensions of strongly π-regular general rings are considered. Two equivalent characterizations on strongly π- regular general rings are provided. It is shown that I is strongly π-regular if and only if, for each x ∈I, x^n =x^n＋1y = zx^n＋1 for n ≥ 1 and y, z ∈ I if and only if every element of I is strongly π-regular. It is also proved that every upper triangular matrix general ring over a strongly π-regular general ring is strongly π-regular and the trivial extension of the strongly π-regular general ring is strongly clean.

Extensions of Symmetric Rings

We first consider properties and basic extensions of symmetric rings. We next argue about the symmetry of some kinds of polynomial rings, and show that if R is a reduced ring then R[x]/（x^n） is a symmetric ring, where （x^n） is the ideal generated by x^n and n is a positive integer. Consequently, we prove that for a right Ore ring R with Q its classical right quotient ring, R is symmetric if and only if Q is symmetric.

Double Frobenius algebras

Some equivalent conditions for double Frobenius algebras to be strict ones are given. Then some examples of （strict or non-strict） double Frobenius algebras are presented. Finally, a sufficient and necessary condition for the trivial extension of a double Frobenius algebra to be a （strict） double Frobenius algebra is given.