Narrow Results

An act A_S is called torsion free if for any a, b ∈ A_S and for any right cancellable element c ∈ S the equality ac = bc implies a = b. In [M. Satyanarayana, Quasi- and weakly-injective S-system, Math. Nachr.71 (1976) 183–190], torsion freeness is considered in a much stronger sense which we call in this paper strong torsion freeness and will characterize monoids by this property of their (cyclic, monocyclic, Rees factor) acts.

In this paper, we investigate the notion of dc-po-flat $S$-posets as the ones for which the associated tensor functors transfer merely down-closed embeddings (embeddings with down-closed images in codomains) to embeddings. We investigate derived flatness notions in regard to dc-po-flatness in parallel with po-flatness notions and give examples to clarify new notions and their implications. As the characterization of flat acts by means of embeddings into cyclic acts, stated by Fleischer, is not valid for $S$-posets, it eventuates in introducing the new notion of cyclical po-flatness, situated strictly between weak po-flatness and po-flatness, though, we express a counterpart characterization for dc-po-flatness. At the end, we expose relationships between some po-flatness properties and regular injectivity.

Let ${𝒜}$ and $𝔄$ be two Banach algebras such that ${𝒜}$ is a Banach $𝔄$-bimodule with the left and right compatible actions of $𝔄$ on ${𝒜}$. Let ${𝒜}⋊𝔄$ be a strongly splitting Banach algebra extension of $𝔄$ by ${𝒜}$. In this paper, we investigate some homological aspects such as injectivity, projectivity and flatness of ${𝒜}⋊𝔄$ and give some necessary and sufficient conditions for injectivity, projectivity and flatness of ${𝒜}⋊𝔄$.