Let R be a commutative ring with identity and T (R) its total quotient ring. We extend the notion of well-centered overring of an integral domain to an arbitrary commutative ring and we investigate the transfer of this property to different extensions of commutative rings in both integral and non-integral cases. Namely in pullbacks and trivial extensions. Our aim is to provide new classes of co...

We deal with the following question: What is the proper way to introduce symmetric difference in orthomodular lattices? Imposing two natural conditions on this operation, six possibilities remain: the two (commutative) normal forms of the symmetric difference in Boolean algebras and four noncommutative terms. It turns out that in many respects the non-commutative forms, though more complex with...

The Maxwell theory on non-commutative spaces has been considered. The non-linear equations of electromagnetic fields on non-commutative spaces were obtained in the compact spin-tensor (quater-nion) form. It was shown that the plane electromagnetic wave is the solution of the system of non-linear wave equations of the second order for the electric and magnetic induction fields. We have found the...

We compute the shear viscosity of the non-commutative N=4 super Yang-Mills quantum field theory at strong coupling using the dual supergravity background. Special interest derives from the fact that the background presents an intrinsic anisotropy in space through the distinction of commutative and non-commutative directions. Despite this anisotropy the analysis exhibits the ubiquitous result η/...

By a commutative term we mean an element of the free commutative groupoid F of infinite rank. For two commutative terms a, b write a ≤ b if b contains a subterm that is a substitution instance of a. With respect to this relation, F is a quasiordered set which becomes an ordered set after the appropriate factorization. We study definability in this ordered set. Among other thing, we prove that e...

Definition 1.1 (Rings). The algebraic structure “ring” R is a set with two binary operations + and ·, respectively named addition and multiplication, satisfying • (R,+) is an abelian group (i.e. a group with commutative addition), • is associative (i.e. 8a, b, c 2 R, (a · b) · c = a · (b · c)) , • and the distributive law holds (i.e. 8a, b, c 2 R, (a+ b) · c = a · c+ b · c, a · (b+ c) = a · b+ ...

We compactify the spaces K(m,n) introduced by Maxim Kontsevich. The initial idea was to construct an L∞ algebra governing the deformations of a (co)associative bialgebra. However, this compactification leads not to a resolution of the PROP of (co)associative bialgebras, but to a new algebraic structure we call here a CROC. It turns out that these constructions are related to the non-commutative...