On the dualization of subdirect embeddings

The classical structure theorems of algebraic systems usually assume some type of finiteness condition. The most common finiteness restriction is a chain condition. Thus the proofs of the fundamental structure and decomposition theorems for lattices have customarily required the ascending chain condition. Moreover, these theorems generally fail to hold in arbitrary lattices. Nevertheless, there are important examples of decomposition theorems for lattices associated with abelian groups and rings in which the ascending chain condition does not hold. These lattices have in common another distinctive property—they are compactly generated. Namely, the lattice is generated by a collection of elements which are finitely dependent in the sense that any such element is contained in the union of a set of lattice elements if and only if it is contained in the union of a finite subset. The compact elements of the lattice of ideals of a ring are the finitely generated ideals. Likewise the compact elements of a lattice of congruence relations are the congruence relations generated by collapsing a finite collection of element pairs. More generally, the lattice of congruence relations and the lattice of subsystems of a universal algebra are compactly generated. Since structure theorems for an algebraic system correspond to decomposition theorems in the lattice of congruence relations, this strongly suggests that compactly generated lattices are the appropriate domain in which to study decomposition theory. Furthermore, since every lattice satisfying the ascending chain condition is trivially compactly generated, it follows that the classical case will be subsumed in the more general theory.

M. E. Sweedler first constructed a universal Hopf algebra of an algebra. It is known that the dual notions to the existing ones play a dominant role in Hopf algebra theory. Yu. I. Manin and D. Tambara introduced the dual notion of Sweedler's construction in separate works. In this paper, we construct a universal algebra for a finite-dimensional Lie-Yamaguti algebra. We demonstrate that this universal algebra possesses a bialgebra structure, leading to a universal coacting Hopf algebra for a finite-dimensional Lie-Yamaguti algebra. Additionally, we develop a representation-theoretic version of our results. As an application, we characterize the automorphism group and classify all abelian group gradings of a finite-dimensional Lie-Yamaguti algebra.

In this paper we give a detailed exposition of some of the results announced in [18]. The primary motivation for this work is Witt's observation [31, Satz 7] that if F is a field of characteristic #72, his ring 1V(F) of classes of anisotropic quadratic forms may be written as Z[G]/K where G is an abelian group of exponent two (actually G =F*/F*2), Z[G] is the integral group ring of G, and K is an ideal of Z[G] generated by elements of the form gl + g2-g3g4 and 1 + g5 with gi in G. Of course these elements can be described more explicitly, but for our purposes the only information we need about K is that any homomorphism of Z[G] -> Z sends K to 0 or to an ideal of the form 2nZ. In this introduction we shall call any ideal of Z[G] with this property admissible. In [26], Pfister proved certain structure theorems for IV(F) usilg his theory of multiplicative forms. His proofs were simplified in [11, 22, 23, 29, 30]. Harrison [11] and Leicht and Lorenz [22] gave important complements to Pfister's results concerning the ideal theory of WV(F). The main goal of our paper is to understand and generalize these structure theorems using standard techniques of commutative algebra. In [28] and [3], Scharlau and Belskii have introduced and studied Witt and Witt-Grothendieck rings for profinite groups. Their definitions generalize the usual notions of Wit.t rings, WV(F), and Witt-Grothendieck rings, K (F), of quadratic forms over fields of characteristic #72. These rings also have the form Z[G]/K with G an abelian group of exponent two and K an admissible ideal of Z[G]. Here we are mainly iinterested in another generalization of 1V(F) and K(F), namely the Witt rings, IV(C, J), and the Witt-Grothendieck rings, K(C,J) of classes of hermitian forms over a connected commutative semilocal ring C with involution J. Since J may be the identity these include the Witt and Witt-Grothendieck rings of classes of symmetric bilinear forms

POLYNOMIAL NORMAL FORMS AND THE EMBEDDING OF POLYNOMIAL ALGEBRAS

We prove in this paper a structure theorem for prime rings whose symmetric ring of quotients has nonzero socle. Then this result is applied to prime rings satisfying a generalized identity, and to prime rings having an alternate involution.

Effective subdirect decomposition: a case study

The title of this book is Universal Algebraic Logic. One reading of this title is: (Universal Algebraic) Logic, meaning a stance of algebraic logic where the algebraic part is universal algebra (since the logic part is universal logic, too). This title indicates the important role of universal algebra in the subject of the present book.

i. A. G. Pinus, "On the relations of epimorphism and embeddability on congruence-distributive varieties," Algebra Logika, 24, NO. 5, 588-607 (1985). 2. A. G. Pinus, Congruence-Modular Varieties of Algebras [in Russianl, Irkutsk State Univ. (1986). 3. A. G. Pinus, "On simple epimorphism skeletons of congruence-distributive varieties," Izv. Vyssh. Uchebn. Zaved., Mat., No. ii, 67-70 (1987). 4. A. G. Pinus, "On coverings in epimorphism skeletons of varieties of algebras," Algebra Logika, 27, No. 3, 316-326 (1988). 5. A. G. Pinus, "ConGruence-distributive varieties of algebras," Progress in Science. Algebra, Geometry, and Toplogy [in Russian], Vol. 26, Bsesoyuz. Inst. Nauch. i Tekh. Informatsii, Moscow (1988), pp. 45-83. 6. A. G. Pinus, "On the number of incomparables in countable epimorphism skeletons of discriminator varieties," Algebra Logika, 2_88, No. 3, 311-323 (1989). 7. G. Gritzer, Universal Algebra, 2nd Ed., Springer-Verlag, Berlin-Heidelberg (1979). 8. S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, New York-Heidelberg-Berlin (1981). H. Werner, Discriminator Algebras, Akademie Verlag, Berlin (1978). G. A. Fraser and A. Horn, "Congruence relations in direct product," Proc. Am. Math. Soc., 26, No. 3, 390-394 (1970). .

Poor modules with no proper poor direct summands

A ring R is called right pseudo-semisimple if every right ideal not isomorphic to R is semisimpie. Rings of this type in which the right socle S splits off additively were characterized; such a ring has S2 = 0. The existence of right pseudo-semisimple rings with zero right singular ideal Z remained open, except for the trivial examples of semisimple rings and principal right ideal domains. In this work we give a complete characterization of right pseudo-semisimple rings with S2 = 0. We also give examples of non-trivial right pseudo-semisimple rings with Z = 0; in fact it is shown that such rings exist as subrings in every infinite-dimensional full linear ring. A structure theorem for non-singular right pseudo-semisimple rings, with homogeneous maximal socle, is given. The general case is still open.

The structure of smooth algebras in Kapranov's framework for noncommutative geometry

A Characterization of All Abelian Groups Whose Lattice of Precompact Group Topologies Represents A Projective Geometry

The purpose of this survey is to provide an introduction to the interesting role which set-theoretic methods have begun to play in the problem of determining the structure and classification of certain classes of abelian groups, in particular, the class of ω1-separable groups, i.e., groups such that every countable subset is contained in a Σ -cyclic summand.

Frame vectors for representations of Abelian groups

Under some very strong set-theoretic hypotheses, hereditarily normal spaces (also referred to as T${}_5$ spaces) that are locally compact and hereditarily collectionwise Hausdorff can have a highly simplified structure. This paper gives a structure theor