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This is a summary of some of the main results in the monograph Faithfully Ordered Rings (Mem. Amer. Math. Soc. 2015), presented by the first author at the ALANT conference, Będlewo, Poland, June 8-13, 2014. The notions involved and the results are stated in detail, the techniques employed briefly outlined, but proofs are omitted. We focus on those aspects of the cited monograph concerning (diagonal) quadratic forms over preordered rings.

Family algebras.

Kirillov, A.A. (2000)

Electronic Research Announcements of the American Mathematical Society [electronic only]

Fastgeordnete und geordnete lokale Ringe und ihre geometrische Anwendung

František Machala (1981)

Časopis pro pěstování matematiky

FC-modules with an application to cotorsion pairs

Yonghua Guo (2009)

Commentationes Mathematicae Universitatis Carolinae

Let $R$ be a ring. A left $R$ -module $M$ is called an FC-module if $M^{+} = {Hom}_{ℤ} (M, ℚ / ℤ)$ is a flat right $R$ -module. In this paper, some homological properties of FC-modules are given. Let $n$ be a nonnegative integer and ${ℱ𝒞}_{n}$ the class of all left $R$ -modules $M$ such that the flat dimension of $M^{+}$ is less than or equal to $n$ . It is shown that $(^{⊥} ({ℱ𝒞}_{n}^{⊥}), {ℱ𝒞}_{n}^{⊥})$ is a complete cotorsion pair and if $R$ is a ring such that $fd ({(_{R} R)}^{+}) \leq n$ and ${ℱ𝒞}_{n}$ is closed under direct sums, then $({ℱ𝒞}_{n}, {ℱ𝒞}_{n}^{⊥})$ is a perfect cotorsion pair. In particular, some known results are obtained as corollaries....

Fiber products and Morita duality for commutative rings

Alberto Facchini (1982)

Rendiconti del Seminario Matematico della Università di Padova

Field-Dependent Homological Behavior of finite dimensional algebras.

Birge Zimmermann Huisgen (1994)

Manuscripta mathematica

Filial rings

Ehrlich, Gertrude (1983/1984)

Portugaliae mathematica

Filtered vector spaces (abstract)

Vlastimil Dlab (1974/1975)

Séminaire Dubreil. Algèbre et théorie des nombres

Filtrations, Stable Clifford Theory and Group-Graded Rings and Modules.

Morton E. Harris (1987)

Mathematische Zeitschrift

Finding a cluster-tilting object for a representation finite cluster-tilted algebra

M. A. Bertani-Økland, S. Oppermann, A. Wrålsen (2010)

Colloquium Mathematicae

We provide a technique to find a cluster-tilting object having a given cluster-tilted algebra as endomorphism ring in the finite type case.

Finite associative rings

R. Raghavendran (1969)

Compositio Mathematica

Finite automata and algebraic extensions of function fields

Kiran S. Kedlaya (2006)

Journal de Théorie des Nombres de Bordeaux

We give an automata-theoretic description of the algebraic closure of the rational function field $𝔽_{q} (t)$ over a finite field $𝔽_{q}$ , generalizing a result of Christol. The description occurs within the Hahn-Mal’cev-Neumann field of “generalized power series” over $𝔽_{q}$ . In passing, we obtain a characterization of well-ordered sets of rational numbers whose base $p$ expansions are generated by a finite automaton, and exhibit some techniques for computing in the algebraic closure; these include an adaptation to positive...

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$f$ -radical extensions of rings

Factors of Wild, Concealed Algebras.

Fair-sized projective modules

Faithful linear representations of bands.

Faithfully quadratic rings - a summary of results