相关论文: On the order theory for $\mathcal{C}^\infty$-reduc…

We develop a general ring theory in the o-minimal setting culminating in a description of all the definable rings in an arbitrary o-minimal structure. We show that every definably connected ring with non-trivial multiplication defines an…

A general method of producing correspondences and spectral categories out of symmetric ring objects in general categories is given. As an application, stable homotopy theory of spectra $SH$ is recovered from modules over a commutative…

This is an introduction to rings and fields, written for a quarter-long undergraduate course. It includes the basic properties of ideals, modules, algebras and polynomials, the constructions of ring extensions and finite fields, some…

This paper presents an extension of the concept of NR-clean introduced in [12] to graded ring theory. We define and explore graded NR-clean rings, which generalize the class of graded U-nil clean previously studied in [15]. We provide…

Genuine equivariant homotopy theory is equipped with a multitude of coherently commutative multiplication structures generalizing the classical notion of an $\mathbb{E}_\infty$-algebra. In this paper we study the…

A ringoid is a set with two binary operations that are linked by the distributive laws. We study special classes of ringoids that are congruence-simple or ideal-simple. In particular, we examine generalised parasemifields and…

Using an idea due to R.Thomason, we define a "homology theory" on the category of rings which satisfies excision, exactness, homotopy (in the algebraic sense) and periodicity of order 4. For regular noetherian rings, we find P. Balmer's…

This is a survey of the author's paper arXiv:1001.0023 on "Algebraic Geometry over C-infinity rings". If X is a smooth manifold then the R-algebra C^\infty(X) of smooth functions c : X --> R is a "C-infinity ring". That is, for each smooth…

We view strict ring spectra as generalized rings. The study of their algebraic K-theory is motivated by its applications to the automorphism groups of compact manifolds. Partial calculations of algebraic K-theory for the sphere spectrum are…

We present equivalent conditions of reverse order law for the $(b, c)$-inverse $(aw)^{(b,c)}=w^{(b,s)}a^{(t,c)}$ to hold in a ring. Also, we study various mixed-type reverse order laws for the $(b,c)$-inverse. As a consequence, we get…

Integral categories were recently developed as a counterpart to differential categories. In particular, integral categories come equipped with an integration operator, known as an integral transformation, whose axioms generalize the basic…

Let B be a ring and $A=B[X,Y]/(aX^2+bXY+cY^2-1)$ where $a,b,c\in B$. We study the smoothness of A over B, and the regularity of B when B is a ring of algebraic integers.

A low energy string theory should reduce to an ordinary quantum field theory, but in reality the structures of the two are so different as to make the equivalence obscure. The string formalism is more symmetrical between the spacetime and…

In the present paper we develop a small cancellation theory for associative algebras with a basis of invertible elements. Namely, we study quotients of a group algebra of a free group and introduce three axioms for the corresponding…

In this article, we focus on orders in arbitrary number fields, consider their Picard groups and finally obtain ring class fields corresponding to them. The Galois group of the ring class field is isomorphic to the Picard group. As an…

Let $\mathcal{R}$ be a unital ring with involution. The notions of 1MP-inverse and MP1-inverse are extended from $M_{m,n}(\mathbb{C)}$, the set of all $m\times n $ matrices over $\mathbb{C}$, to the set $\mathcal{R}% ^{\dagger}$ of all…

We prove that a vector field on an affine $C^\infty$-scheme Spec(A) has a flow if the $C^\infty$-ring A is finitely generated. If the vector field is complete then the flow is the target map of a groupoid internal to the category of…

We construct a commutative orthogonal $C_2$-ring spectrum, $\mathrm{MSpin}^c_{\mathbb{R}}$, along with a $C_2$-$E_{\infty}$-orientation $\mathrm{MSpin}^c_{\mathbb{R}} \to \mathrm{KU}_{\mathbb{R}}$ of Atiyah's Real K-theory. Further, we…

We apply recent results on the rank of elements of rings to study the structure of generalized corner rings $aRa$, where $R$ is a unital ring and $a$ an element of $R$. We give a complete description of the structure of $aRa$ when $a^2$ has…

We show that the reduced groupoid C*-algebras of continuous fields of \'etale groupoids satisfying the rapid decay property yield continuous fields of C*-algebras. This establishes a new sufficient criterion that applies in the non-amenable…