Non-Commutative Ring Theory Research Papers (original) (raw)

We give an equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations. A consequence of this specification is that 0 −1 = 0, an interesting... more

We give an equational specification of the field operations on the rational numbers under initial algebra semantics using just total field operations and 12 equations. A consequence of this specification is that 0 −1 = 0, an interesting equation consistent with the ring axioms and many properties of division. The existence of an equational specification of the rationals without hidden functions was an open question. We also give an axiomatic examination of the divisibility operator, from which some interesting new axioms emerge along with equational specifications of algebras of rationals, including one with the modulus function. Finally, we state some open problems, including: Does there exist an equational specification of the field operations on the rationals without hidden functions that is a complete term rewriting system?

We consider the abelian group PTPTPT generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semi-orthogonal decompositions of corresponding triangulated categories. We introduce an operation of... more

We consider the abelian group PTPTPT generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semi-orthogonal decompositions of corresponding triangulated categories. We introduce an operation of "multiplication" bullet\bulletbullet on the collection of DG categories which makes this abelian group into a commutative ring. A few applications are considered: representability of "standard" functors between derived categories of coherent sheaves on smooth projective varieties and a construction of an interesting motivic measure.

Abstract. In this paper we consider categories over a commutative ring pro-vided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that... more

Abstract. In this paper we consider categories over a commutative ring pro-vided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the ...

A contractive n-tuple A = (A1,…,An ) has a minimal joint isometric dilation S = (S 1,…,S n) where the Si ’s are isometries with pairwise orthogonal ranges. This determines a representation of the Cuntz-Toeplitz algebra. When A acts on a... more

A contractive n-tuple A = (A1,…,An ) has a minimal joint isometric dilation S = (S 1,…,S n) where the Si ’s are isometries with pairwise orthogonal ranges. This determines a representation of the Cuntz-Toeplitz algebra. When A acts on a finite dimensional space, the wot-closed nonself-adjoint algebra generated by S is completely described in terms of the properties of A. This provides complete unitary invariants for the corresponding representations. In addition, we show that the algebra is always hyper-reflexive. In the last section, we describe similarity invariants. In particular, an n-tuple B of d × d matrices is similar to an irreducible n-tuple A if and only if a certain finite set of polynomials vanish on B.

The theory of Lambda-rings, in the sense of Grothendieck's Riemann-Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z of integers to produce... more

The theory of Lambda-rings, in the sense of Grothendieck's Riemann-Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z of integers to produce Lambda-algebraic geometry. We show that Lambda-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than Z and that it has many properties predicted for algebraic geometry over the mythical field with one element. Moreover, it does this is a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry.

We develop and compare two approaches to the theory of Thom spectra. The first involves a rigidified model of A-infinity and E-infinity spaces. Our second approach is via infinity categories. In order to compare these approaches to one... more

We develop and compare two approaches to the theory of Thom spectra. The first involves a rigidified model of A-infinity and E-infinity spaces. Our second approach is via infinity categories. In order to compare these approaches to one another and to the classical theory, we characterize the Thom spectrum functor from the perspective of Morita theory.

There is an increasing body of evidence that prime near-rings with derivations have ring like behavior, indeed, there are several results (see for example (1), (2), (3), (4), (5) and (8)) asserting that the existence of a... more

There is an increasing body of evidence that prime near-rings with derivations have ring like behavior, indeed, there are several results (see for example (1), (2), (3), (4), (5) and (8)) asserting that the existence of a suitably-constrained derivation on a prime near-ring forces the near-ring to be a ring. It is our purpose to explore further this ring like

Let R be a commutative ring with non-zero unity. The concept of the graph of the zero-divisors of R was first introduced by Beck [4], where he was mainly interested in colorings. In his work all elements of the ring were vertices of the... more

Let R be a commutative ring with non-zero unity. The concept of the graph of the zero-divisors of R was first introduced by Beck [4], where he was mainly interested in colorings. In his work all elements of the ring were vertices of the graph. This investigation of colorings of a ...