In general, a product has a different theory to that of the structures in the set Mi. For example a product of fields is just a ring. Note also that if φ is an atomic formula then ∏ iMi |= φ((ai )i, · · · , (ai )i) iff {i ∈ I : Mi |= φ(ai , · · · , ai )} = I. In particular this shows that if J ⊆ I then the restiction function (ai)i∈I → (ai)i∈J is a surjective homomorphism from ∏ i∈IMi to ∏ i∈JM...

In this paper, we prove the generalized Hyers–Ulam stability of homomorphisms and (θ, φ)-derivations on a ring R into a Banach R-bimodule M.

In the study of hereditary Noetherian rings, it is clear that hereditary Noetherian prime rings will play a central role (see, for example, [12]). Here we study the (two-sided) ideals of an hereditary Xoetherian prime ring and, as a consequence, ascertain the structure of factor rings and torsion modules. The torsion theory represents a generalization of similar results about Dedekind prime rin...

If G is a finite abelian group, R is a principal ideal domain with field of quotients an algebraic number field K which splits G, and if A is a Hopf algebra order in KG, then the Grothendieck ring of the category of finitely generated A-modules is isomorphic to the Grothendieck ring of the category of finitely generated RG-modules. The Grothendieck group a£ of the category of finitely generated...

let φ be a w  -continuous homomorphism from a dual banach algebra to c. the notion of φ-connes amenability is studied and some characterizations is given. a type of diagonal for dual banach algebras is dened. it is proved that the existence of such a diagonal is equivalent to φ-connes amenability. it is also shown that φ-connes amenability is equivalent to so-called φ-splitting of a certain s...

This talk will provide a snapshot of contemporary commutative algebra. In classical commutative algebra and algebraic number theory, the Dedekind domains are the most important class of rings. Modern commutative algebra studies numerous generalizations of the Dedekind domains in attempts to generalize results of algebraic number theory. This talk will introduce a few important generalizations o...

Interpretation of Predicates In this section we give an example of the abstract interpretation of a predicate which has a conjunction. The predicate tests if the parameters S and R which both naturally carry the structure ordered set actually both carry the structure ring. The predicate is: 2The source of the code is found in the file “EXPR2.spad” in OpenAxiom [Ope09] 5.4. ABSTRACT INTERPRETATI...

Let F be a field with char(F ) 6= 2. We prove Vishik’s conjecture stating that for any positive integer n and any anisotropic quadratic form φ over F lying in the n-th power I of the fundamental ideal I of the Witt ring of F , either dim(φ) ≥ 2 or dim(φ) = 2 − 2 for some i ∈ {1, 2, . . . , n+ 1}. This provides a complete solution of an old-standing problem in the algebraic theory of quadratic f...