These rings admit numerous algebraic and geometric realizations, but one of the historically first constructions, which dates back to the work of Steinitz in 1900 and later completed by Hall, was given in terms of what is now called the classical Hall algebra H (see [Ma], Chapter III ). This algebra has a basis consisting of isomorphism classes of abelian q-groups, where q is a fixed prime powe...

Methods from homological algebra [16] play a more and more important role in the study of multidimensional linear systems [15], [14], [6]. The use of modules allows an algebraic treatment of linear systems which is independent of their presentations by systems of equations. The type of linear system (ordinary/partial differential equations, timedelay systems, discrete systems. . . ) is encoded ...

A ternary ring of operators is an “off-diagonal corner” of a C∗-algebra and the predual of a ternary ring of operators (if it exists) is of the form pR∗q for some von Neumann algebra R and projections p and q in R. In this paper, we prove that a subspace of the predual of a ternary ring of operators is completely 1-complemented if and only if it is completely isometrically isomorphic to the pre...

The purpose of this paper is to study the tropical algebra – the algebra over the tropical semi-ring. We start by introducing a third approach to arithmetic over the max-plus semi-ring which generalizes the two other concepts in use. Regarding this new arithmetic, matters of tropical matrices are discussed and the properties of these matrices are studied. These are the preceding phases toward t...

We explore two constructions in homotopy category with algebraic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an extension of the algebraic theory to include the case when M is not necessarily a progenerator. Our approach is complementary to recent work of Dwyer & Greenlee...

In this paper, we study the maximal ideals in a commutative ring of bicomplex numbers and then describe algebra. We found that kernel nonzero multiplicative BC-linear functional Banach algebra need not be ideal. Finally, introduce notion division generalize Gelfand–Mazur theorem for

We explore two constructions in homotopy category with algebraic precursors in the theory of noncommutative rings and homological algebra, namely the Hochschild cohomology of ring spectra and Morita theory. The present paper provides an extension of the algebraic theory to include the case when M is not necessarily a progenerator. Our approach is complementary to recent work of Dwyer and Greenl...