A ring is called quasi-Frobenius if it is right or left selfinjective, and right or left artinian (all four combinations are equivalent). The study of these rings grew out of the theory of representations of a finite group as a group of matrices over a field, and the subject is intimately related to duality, the duality from right to left modules induced by the hom functor and the duality related to annihilators. The present extent of the theory is vast, and this book makes no attempt to be encyclopedic; instead it provides an elementary, self-contained account of the basic facts about these rings at a level allowing researchers and graduate students to gain entry to the field.

/pdf/quasi-frobenius-rings-2qqf5oyqi3.pdf

Quasi-Frobenius Rings

/pdf/separate-continuity-and-joint-continuity-1wlhvewj3i.pdf

Separate continuity and joint continuity

We prove a number of results concerning Armendariz rings and Gaussian rings. Recall that a (commutative) ring R is (Gaussian) Armendariz if for two polynomials f,g∈R[X] (the ideal of R generated by the coefficients of f g is the product of the ideals generated by the coefficients of f and g) fg = 0 implies a i b j=0 for each coefficient a i of f and b j of g. A number of examples of Armendariz rings are given. We show that R Armendariz implies that R[X] is Armendariz and that for R von Neumann regularR is Armendariz if and only if R is reduced. We show that R is Gaussian if and only if each homomorphic image of R is Armendariz. Characterizations of when R[X] and R[X] are Gaussian are given.

Armendariz rings and gaussian rings

QF-3 rings

If τ is a projection of a closed convex polyhedron P onto a convex polyhedron Q, then a lifting of Q into P is defined to be a single-valued inverse τ* of τ such that τ*(Q) is the union of closed faces of P The main result of this paper, designated the Lifting Theorem, asserts that there always exists a lifting r*, provided only that there exists at least one face of P on which τ acts one-to-one The lifting theorem represents a unifying generalization of a number of results in the theory of convex polyhedra and should prove useful as an investigative as well as a conceptual tool In the course of the proof, a special case of the Lifting Theorem is translated into linear programming terms and stated as the Basis Decomposition Theorem, which summarizes the behavior of a linear program as a function of its right-hand sides In particular, the fact that a lifting is necessarily a piecewise linear homeomorphism is reflected in the Basis Decomposition Theorem as the observation that the optimal solution of a linear program can always be chosen as a continuous function of the right-hand sides

/pdf/lifting-projections-of-convex-polyhedra-6idgfaibdg.pdf

Lifting projections of convex polyhedra.

Conference on Commutative Algebra

Let R be a ring with minimum condition on left or right ideals. It is shown that R is a QF-Z ring if and only if each finitely generated submodule of the injective hull of R, regarded as a left ϋί-module, is torsionless. The same approach yields a simplified proof that R is quasi-Frobenius if and only if every finitely generated left i?-module is torsionless. A ring with identity is called a left QF-Z ring if it has a (unique) minimal faithful left module, and a QF-Z ring means a ring which is both left and right QF-Z. This class of rings originated with Thrall [9] as a generalization of quasi-Frobeni us or QF algebras and has been studied extensively in recent years. Quasi-Frobenius rings have many interesting characterizations and in most instances there exists an analogous characterization of QF-Z rings at least in the case of rings with minimum condition and often for a much larger class of rings. It is well known that a ring with minimum condition on left or right ideals is a left QF-Z ring if and only if the injective hull E{BR) of the ring R regarded as a left iϋ-module is protective. Moreover, in this case R is a QF-Z ring (cf. [6] and [8]). For semi-primary or perfect rings; however, the situation is somewhat different. Namely, a perfect ring is a left QF-Z ring if and only if E(RR) is torsionless. A module is called torsionless if it can be embedded in a direct product of copies of the ring regarded as a module over itself. In this case E(RR) need not be protective and R need not be right QF-Z (cf. [3] and [8]). However, a perfect ring is QF-Z if and only if both E(RR) and E(RR) are protective (see [8]). In this note, it is shown that if R is left perfect ring, E(RR) is protective if and only if each finitely generated submodule of E{RR) can be embedded in a free i?-module. For a ring with minimum condition on left or right ideals this latter condition is equivalent to each finitely generated submodule of E{BR) being torsionless. Thus in that case QF-Z rings may be characterized by this weaker condition. The technique of proof also yields a much simplified proof of a characterization of QF rings given by the present author in [7]. Namely, a ring with minimum condition on left or right ideals is QF if and only if each finitely generated left module is torsionless. Indeed, the characterization of QF-Z rings given here may be regarded as the analog of that result.

/pdf/a-characterizati-on-of-qf-3-rings-y7mc2tywty.pdf

A characterizati on of qf-3 rings

/pdf/qf-3-rings-with-zero-singular-ideal-t85sm78wng.pdf

QF-3 rings with zero singular ideal

This paper investigates two related problems. The first is to describe the double centralizer of an arbitrary projective right iϋ-modiile. This proves to be the ring of left quotients of R with respect to a certain canonical hereditary torsion class of left ϋί-modules determined by the projective module. The second is to determine the relationship between rings of left quotients of R and S, where S is the endomorphism ring of a finitely generated projective right ϋί-module PR. It is shown that there exists an inclusion-preserving, one-to-one correspondence between hereditary torsion classes (or localizing subcategories) of left ^-modules and hereditary torsion classes of left E-modules which contain the canonical torsion class determined by PR. If QR and Qs are rings of left quotients with respect to corresponding classes, then P($$RQR is a finitely generated projective right Q^-module with Qs as its Q#-endomorphism ring. Necessary and sufficient conditions are obtained for the maximal rings of left quotients to be related in this manner. In particular, this occurs when PR is a faithful .R-module and R is either a semi-prime ring or a ring with zero left singular ideal. The situation considered includes the case where S is an arbitrary ring, SP is a left ^-generator, and R is the Sendomorphism ring of SP When SP is a projective left Sgenerator, the maximal rings of left quotients of R and S are related in the manner considered above.

/pdf/rings-of-quotients-of-endomorphism-rings-of-projective-1yfxl1lypv.pdf

Rings of quotients of endomorphism rings of projective modules.

Rings with the principal extension property

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