### A remark on power series rings.

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.

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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.

Let R be a parabolic subgroup in $G{L}_{n}$. It acts on its unipotent radical ${R}_{u}$ and on any unipotent normal subgroup U via conjugation. Let Λ be the path algebra ${k}_{t}$ of a directed Dynkin quiver of type with t vertices and B a subbimodule of the radical of Λ viewed as a Λ-bimodule. Each parabolic subgroup R is the group of automorphisms of an algebra Λ(d), which is Morita equivalent to Λ. The action of R on U can be described using matrices over the bimodule B. The advantage of this description is that each...

Let $f:A\to B$ and $g:A\to C$ be two ring homomorphisms and let $J$ and ${J}^{\text{'}}$ be ideals of $B$ and $C$, respectively, such that ${f}^{-1}\left(J\right)={g}^{-1}\left({J}^{\text{'}}\right)$. In this paper, we investigate the transfer of the notions of Gaussian and Prüfer rings to the bi-amalgamation of $A$ with $(B,C)$ along $(J,{J}^{\text{'}})$ with respect to $(f,g)$ (denoted by $A{\bowtie}^{f,g}(J,{J}^{\text{'}})),$ introduced and studied by S. Kabbaj, K. Louartiti and M. Tamekkante in 2013. Our results recover well known results on amalgamations in C. A. Finocchiaro (2014) and generate new original examples of rings possessing these properties.

Let Λ be a directed finite-dimensional algebra over a field k, and let B be an upper triangular bimodule over Λ. Then we show that the category of B-matrices mat B admits a projective generator P whose endomorphism algebra End P is quasi-hereditary. If A denotes the opposite algebra of End P, then the functor Hom(P,-) induces an equivalence between mat B and the category ℱ(Δ) of Δ-filtered A-modules. Moreover, any quasi-hereditary algebra whose category of Δ-filtered modules is equivalent to mat...

It is well known that the monoid ring of the free product of a free group and a free monoid over a skew field is a fir. We give a proof of this fact that is more direct than the proof in the literature.

This is a description of some different approaches which have been taken to the problem of generalizing the algebraic closure of a field. Work surveyed is by Enoch and Hochster (commutative algebra), Raphael (categories and rings of quotients), Borho (the polynomial approach), and Carson (logic).Later work and applications are given.

Let $\mathcal{T}$ be a weak torsion class of left $R$-modules and $n$ a positive integer. A left $R$-module $M$ is called $(\mathcal{T},n)$-injective if ${\mathrm{Ext}}_{R}^{n}(C,M)=0$ for each $(\mathcal{T},n+1)$-presented left $R$-module $C$; a right $R$-module $M$ is called $(\mathcal{T},n)$-flat if ${\mathrm{Tor}}_{n}^{R}(M,C)=0$ for each $(\mathcal{T},n+1)$-presented left $R$-module $C$; a left $R$-module $M$ is called $(\mathcal{T},n)$-projective if ${\mathrm{Ext}}_{R}^{n}(M,N)=0$ for each $(\mathcal{T},n)$-injective left $R$-module $N$; the ring $R$ is called strongly $(\mathcal{T},n)$-coherent if whenever $0\to K\to P\to C\to 0$ is exact, where $C$ is $(\mathcal{T},n+1)$-presented and $P$ is finitely generated projective, then $K$ is $(\mathcal{T},n)$-projective; the ring $R$ is called $(\mathcal{T},n)$-semihereditary...