for two algebras $a$ and $b$, a linear map $t:a longrightarrow b$ is called separating, if $xcdot y=0$ implies $txcdot ty=0$ for all $x,yin a$. the general form and the automatic continuity of separating maps between various banach algebras have been studied extensively. in this paper, we first extend the notion of separating map for module case and then we give a description of a linear separa...

We show that if X is a toric scheme over a regular commutative ring k then the direct limit of the K-groups of X taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was previously known for regular commutative rings containing a field. The affine case of our result was conjectured by Gubeladze. We prove analogous results when k is replaced by an appropri...

Given a collection P = {p1(x1, . . . , x2k2), . . . , pk2(x1, . . . , x2k2)} of k commutative polynomials in 2k variables, the objective is to find a condensed representation for these polynomials in terms of a single non-commutative polynomial p(X,Y ) in two k × k matrix variables X and Y . Algorithms that will generically determine whether the given family P has a non-commutative representati...

In this paper, we shall consider the Lie algebra of column-finite infinite matrices indexed by positive integers N. We describe lattice its ideals for an arbitrary field K and study derivations over any commutative, unital ring R.

Molodtsov's soft set theory provides a general mathematical framework for dealing with uncertainty. The concepts of (M, N)-SI implicative (Boolean) filters of BL-algebras are introduced. Some good examples are explored. The relationships between (M, N)-SI filters and (M, N)-SI implicative filters are discussed. Some properties of (M, N)-SI implicative (Boolean) filters are investigated. In part...

In this paper, the notation of interior KU-algebra and some related definitions are introduced. Various characterisations for have been proved. The concept positive implicative prove results regarding also presented. addition, KU-ideals several properties these ideals

Hyper-positive real, matrix-valued, rational functions are associated with absolute stability (the Lurie problem). Here, quantitative subsets of functions, related through nested inclusions, introduced. Structurally, this family turns out to be matrix-convex and closed under inversion. A state-space characterization these a corresponding Kalman-Yakubovich-Popov Lemma, is given. Technically, the...

Let $R$ be a non-commutative ring with unity. The commuting graph of $R$ denoted by $Gamma(R)$, is a graph with vertex set $RZ(R)$ and two vertices $a$ and $b$ are adjacent iff $ab=ba$. In this paper, we consider the commuting graph of non-commutative rings of order pq and $p^2q$ with Z(R) = 0 and non-commutative rings with unity of order $p^3q$. It is proved that $C_R(a)$ is a commutative ring...