Pham Ngoc Ánh

Gauss' Lemma and valuation theory

Abstract Gauss’ lemma is not only critically important in showing that polynomial rings over unique factorization domains retain unique factorization; it unifies valuation theory. It figures centrally in Krull’s classical construction of valued fields with pre-described value groups, and plays a crucial role in our new short proof of the Ohm-Jaffard-Kaplansky theorem on Bezout domains with given lattice-ordered abelian groups. Furthermore, Eisenstein’s criterion on the irreducibility of polynomials as well as Chao’s beautiful extension of Eisenstein’s criterion over arbitrary domains, in particular over Dedekind domains, are also obvious consequences of Gauss’ lemma. We conclude with a new result which provides a Gauss’ lemma for Hermite rings.

Morita Duality, Linear Compactness and AB5: A Survey

The idea of duality occurs in many parts of mathematics. Its earliest appearance (some 150 years ago) was in projective geometry where it permitted to halve the number of theorems to be proved. It also played a most important role in analysis and topology, for example in Banach spaces and algebraic topology. In algebra, beginning with the duality of finite abelian groups as well as that of finite dimensional vector spaces, its role has been no less important. As one of the most natural generalizations of the latter for modules, Morita duality has become an important part of textbooks or a topic of lecture notes on rings and modules (cf. Anderson and Fuller [2], Cohn [23], Faith [34], Kasch [47] and Xue [84]). Since the results concerning Morita duality are very numerous and there is the excellent survey paper of Muller [70], it cannot be the purpose of this paper to make a complete report on all aspects and results relating to Morita duality. In the present survey we intend to highlight the basic ideas and observations of Morita duality (in particular, we will stress the importance of linear compactness and Grothendieck’s condition AB5*) and to show how much of this theory can be deduced from more general results about Baer duality, as well as from results of a purely category-theoretical nature. Furthermore, we will touch upon aspects of duality in some important special classes of rings, however, without aiming at completeness.

SELFDUALITIES OF SERIAL RINGS, REVISITED

A description is given of serial rings whose maximal quotient rings are quasi-Frobenius (QF). Every serial ring is a factor of a serial ring whose maximal quotient ring is a QF-ring. This result is used to give a new, conceptual proof for the selfduality of serial rings emphasising the importance of weakly symmetric rings.

Divisibility Theory of Arithmetical Rings with One Minimal Prime Ideal

Continuing the study of divisibility theory of arithmetical rings started in [1] and [2], we show that the divisibility theory of arithmetical rings with one minimal prime ideal is axiomatizable as Bezout monoids with one minimal m-prime filter. In particular, every Bezout monoid with one minimal m-prime filter is order-isomorphic to the partially ordered monoid with respect to inverse inclusion, of principal ideals in a Bezout ring with a smallest prime ideal. Although this result can be considered as a satisfactory answer to the divisibility theory of both semihereditary domains and valuation rings, the general representation theory of Bezout monoids is still open.

A note on self-dual modules and Dedekind rings

Abstract It is shown that a commutative noetherian ring is a finite direct sum of Dedekind rings and artinian uniserial rings if and only if every module of finite length is selfdual. A module of finite length is said to be selfdual if it is isomorphic to its dual with respect to the minimal injective cogenerator.