All rings are commutative with identity and all modules are unital. Let R be a ring, M an R-module and R(M), the idealization of M . Homogeneuous ideals of R(M) have the form I(+)N where I is an ideal of R, N a submodule of M such that IM ⊆ N . In particular, [N : M ] (+)N is a homogeneous ideal of R(M). The purpose of this paper is to investigate how properties of the ideal [N : M ](+)N are re...

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It is argued that the geometric dual of a noncommutative operator algebra represents a notion of quantum state space which differs from existing notions by representing observables as maps from states to outcomes rather than from states to distributions on outcomes. A program of solving for an explicitly geometric manifestation of quantum state space by adapting the spectral presheaf, a constru...

In this paper, we analyze the security of cryptosystems using short generators over ideal lattices such as candidate multilinear maps by Garg, Gentry and Halevi and fully homomorphic encryption by Smart and Vercauteren. Our approach is based on a recent work by Cramer, Ducas, Peikert and Regev on analysis of recovering a short generator of an ideal in the q-th cyclotomic field for a prime power...

This paper studies the representations of a non-negative polynomial f on a non-compact semi-algebraic set K modulo its critical ideal. Under the assumption that the semi-algebraic set K is regular and f satisfies the boundary Hessian conditions (BHC) at each zero of f in K. We show that f can be represented as a sum of squares (SOS) of real polynomials modulo its critical ideal if f ≥ 0 on K. P...

Emil Kraepelin's nosology has been reinvented, for better or worse. In the United States, the rise of the neo-Kraepelinian nosology of DSM-III resuscitated Kraepelin's work but also differed from many of his ideas, especially his overtly biological ontology. This neo-Kraepelinian system has led to concerns regarding overdiagnosis of psychiatric syndromes ("nosologomania") and perhaps scientific...

let a be a unital r-algebra and m be a unital a-bimodule. it is shown that every jordan derivation of the trivial extension of a by m, under some conditions, is the sum of a derivation and an antiderivation.

Generalizing a concept of Lipshitz, Ozsváth and Thurston from Bordered Floer homology, we define D-structures on algebras of unital operads. This construction gives rise to an equivalence of derived categories, which can be thought of as a unital version of Koszul duality using non-unital Quillen homology, even though the non-unital Quillen homology of unital objects is zero.

‎let $omega$ be a class of unital‎ ‎$c^*$-algebras‎. ‎we introduce the notion of a local tracial $omega$-algebra‎. ‎let $a$ be an $alpha$-simple unital local tracial $omega$-algebra‎. ‎suppose that $alpha:gto $aut($a$) is an action of a finite group $g$ on $a$‎ ‎which has a certain non-simple tracial rokhlin property‎. ‎then the crossed product algebra‎ ‎$c^*(g,a,alpha)$ is a unital local traci...

Let R be a (nonzero commutative unital) ring. If I and J are ideals of R such that R/I ⊕R/J is a cyclic R-module, then I + J = R. The rings R such that R/I ⊕R/J is a cyclic R-module for all distinct nonzero proper ideals I and J of R are the following three types of principal ideal rings: fields, rings isomorphic to K ×L for the fields K and L, and special principal ideal rings (R,M) such thatM...