### A Class of Balanced Non-Uniserial Rings.

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Applying the classical work of Nakayama [Ann. of Math. 40 (1939)], we exhibit a general form of non-Frobenius self-injective finite-dimensional algebras over a field.

Let $R$ be an associative ring with identity and let $J\left(R\right)$ denote the Jacobson radical of $R$. $R$ is said to be semilocal if $R/J\left(R\right)$ is Artinian. In this paper we give necessary and sufficient conditions for the group ring $RG$, where $G$ is an abelian group, to be semilocal.

A trivializability principle for local rings is described which leads to a form of weak algorithm for local semifirs with a finitely generated maximal ideal whose powers meet in zero.

A ring A is called a chain ring if it is a local, both sided artinian, principal ideal ring. Let R be a commutative chain ring. Let A be a faithful R-algebra which is a chain ring such that Ā = A/J(A) is a separable field extension of R̅ = R/J(R). It follows from a recent result by Alkhamees and Singh that A has a commutative R-subalgebra R₀ which is a chain ring such that A = R₀ + J(A) and R₀ ∩ J(A) = J(R₀) = J(R)R₀. The structure of A in terms of a skew polynomial ring over R₀ is determined.

In this paper, we introduce a subclass of strongly clean rings. Let $R$ be a ring with identity, $J$ be the Jacobson radical of $R$, and let ${J}^{\#}$ denote the set of all elements of $R$ which are nilpotent in $R/J$. An element $a\in R$ is called very ${J}^{\#}$-clean provided that there exists an idempotent $e\in R$ such that $ae=ea$ and $a-e$ or $a+e$ is an element of ${J}^{\#}$. A ring $R$ is said to be very ${J}^{\#}$-clean in case every element in $R$ is very ${J}^{\#}$-clean. We prove that every very ${J}^{\#}$-clean ring is strongly $\pi $-rad clean and has stable range one. It is shown...

Addendum to the author's article "Rings whose modules have maximal submodules", which appeared in Publicacions Matemàtiques 39, 1 (1995), 201-214.

We list some typos and minor correction that in no way affect the main results of Rings with zero intersection property on annihilators: Zip rings (Publicacions Matemàtiques 33, 2 (1989), pp. 329-338), e.g., nothing stated in the abstract is affected.

The study of stable equivalences of finite-dimensional algebras over an algebraically closed field seems to be far from satisfactory results. The importance of problems concerning stable equivalences grew up when derived categories appeared in representation theory of finite-dimensional algebras [8]. The Tachikawa-Wakamatsu result [17] also reveals the importance of these problems in the study of tilting equivalent algebras (compare with [1]). In fact, the result says that if A and B are tilting...