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A Course in Universal Algebra

We are interested in sheaf constructions in model-theory, so an attempt is made to unify and generalize the results to date, namely various forms of the Feferman-Vaught Theorem, positive decidability results, and constructions of model companions. The task is considerably simplified by introducing a new definition of sheaf constructions over Boolean spaces. We start the paper by giving a new formulation in §1 of what sheaf constructions over Boolean spaces are-perhaps an appropriate name for our construction would be 'Boolean product'. In §2 we show that this construction is closely related to the formation of reduced products, in particular the universal Horn class generated by a class is the class of structures which can be embedded into the Boolean products of the given class. A specialized Boolean product operator F is indeed remarkably similar to reduced products, for in §3 it is shown that V preserves a sentence iff it is equivalent to a Horn sentence; and in §4 we take another look at Comer's version of the Feferman and Vaught Theorem (for sheaves) and show that it is the most general of those currently in use. Several model-theoretic results proved by Waszkiewicz and Weglorz for reduced limit powers have natural extensions to Boolean products as one also sees in §4. In §5 we take a construction introduced by Arens and Kaplansky, which we call a filtered Boolean power, and show how to extend a translation introduced by Ershov to such structures-this translation is a considerable simplification of the Feferman and Vaught approach of §4. Then in §6 we prove the result which motivated us to look at the elementary properties of sheaf constructions, namely that countable Boolean products of finitely many finite structures with a finite language have a decidable theory. This generalizes Comer's results for monadic algebras and (xm = x)-rings, and Ershov's results for bounded Boolean powers. The next section contains technical results needed for the study of model Received by the editors September 17, 1976 and, in revised form, February 8, 1977. AMS (MOS) subject classifications (1970). Primary 02H13, 02G05; Secondary 02G10, 02G15, 02G20, 08A20.

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Sheaf constructions and their elementary properties

In this fourth and final volume the author extends Buchberger's Algorithm in three different directions. First, he extends the theory to group rings and other Ore-like extensions, and provides an operative scheme that allows one to set a Buchberger theory over any effective associative ring. Second, he covers similar extensions as tools for discussing parametric polynomial systems, the notion of SAGBI-bases, Grobner bases over invariant rings and Hironaka's theory. Finally, Mora shows how Hilbert's followers - notably Janet, Gunther and Macaulay - anticipated Buchberger's ideas and discusses the most promising recent alternatives by Gerdt (involutive bases) and Faugere (F4 and F5). This comprehensive treatment in four volumes is a significant contribution to algorithmic commutative algebra that will be essential reading for algebraists and algebraic geometers.

Solving Polynomial Equation Systems I

Varieties of algebras

Part I An array of twentieth century associative algebra: Direct product and sums of rings and modules and the structure of fields Introduction to ring theory: Schur's lemma and semisimple rings, prime and primitive rings, nil, prime and Jacobson radicals Direct sum decompositions of projective and injective modules Direct product decompositions of von Neumann regular rings and self-injective rings Direct sums of cyclic modules When injectives are flat: Coherent FP-injective rings Direct decompositions and dual generalizations of Noetherian rings Completely decomposable modules and the Krull-Schmidt-Azumaya theorem Polynomial rings over Vamosian and Kerr rings, valuation rings and Prufer rings Isomorphic polynomial rings and matrix rings Group rings and Maschke's theorem revisited Maximal quotient rings Morita duality and dual rings Krull and global dimensions Polynomial identities and PI-rings Unions of primes, prime avoidance, associated prime ideals, ACC on irreducible ideals and annihilator ideals in commutative rings Dedekind's theorem on the independence of automorphisms revisited Part II Snapshots of some mathematical friends and places: Snapshots of some mathematical friends and places Index to part II (snapshots) Bibliography Register of names Index of terms and authors of theorems

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Rings and Things and a Fine Array of Twentieth Century Associative Algebra

Modules over Commutative Regular Rings

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Modules over Commutative Regular Rings

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A Course in Universal Algebra

Sheaf constructions and their elementary properties

Solving Polynomial Equation Systems I

Varieties of algebras

Rings and Things and a Fine Array of Twentieth Century Associative Algebra

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