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.

We revisit the characterisation of modules over non-unital C∗-algebras analogous to modules of sections of vector bundles. A fullness condition on the associated multiplier module characterises a class of modules which closely mirror the commutative case. We also investigate the multiplier-module construction in the context of bi-Hilbertian bimodules, particularly those of finite numerical inde...

Throughout the text, let R be a commutative ring with identity (often called a commutative unital ring) and with multiplicative group of units R∗. Likewise, let f(x) = a0x +a1x n−1+ · · ·+an−1x+an be a polynomial of the variable x over R such that a0 ∈ R∗. Traditionally, R[x] is the ring of all polynomials of x over R; thereby f(x) ∈ R[x]. For an arbitrary but fixed element α, suppose f(x) is t...

Basic Definition: Let R be a commutative ring with 1. A (unital) R-module is an abelian group M together with a operation R ×M → M , usually just written as rv when r ∈ R and v ∈ M . This operation is called scaling . The scaling operation satisfies the following conditions. 1. 1v = v for all v ∈M . 2. (rs)v = r(sv) for all r, s ∈ R and all v ∈M . 3. (r + s)v = rv + sv for all r, s ∈ R and all ...

We consider algebras over a field k of characteristic zero. The article is concerned with the isomorphism of graded vectorspaces H(gl(A)) ∼ → ∧(HC(A)[−1]) between the Lie algebra homology of matrices and the free graded commutative algebra on the cyclic homology of the k-algebra A, shifted down one degree. For unital algebras this isomorphism is a classical result obtained by Loday and Quillen ...

let $r$ be a commutative ring with identity and $m$ be a finitely generated unital $r$-module. in this paper, first we give necessary and sufficient conditions that a finitely generated module to be a multiplication module. moreover, we investigate some conditions which imply that the module $m$ is the direct sum of some cyclic modules and free modules. then some properties of fitting ideals of...

Let A and B be unital semi-simple commutative Banach algebras. In this paper we study two-variable polynomials p which satisfy the following property: a map T from A onto B such that the equality σ(p(Tf, T g)) = σ(p(f, g)), f, g ∈ A holds is an algebra isomorphism.

We consider extensions of unital commutative rings. We define an extension R ↪→ S to be a p-extension if every principally generated ideal of S is generated by an element of R. Examples are plentiful and localizations of regular multiplicative sets are p-extensions. We develop the theory of pextensions.

The C*-algebra of bounded operators on the separable Hilbert space cannot be mapped to a W*-algebra in such a way that each unital commutative C*-subalgebra C(X) factors normally through `∞(X). Consequently, there is no faithful functor discretizing C*-algebras to W*-algebras this way.

Let R be a commutative ring with identity and M be a unital R-module. Then M is called a multiplication module provided for every submodule N of M there exists an ideal I of R such that N = IM. Our objective is to investigate properties of prime and semiprime submodules of multiplication modules. Mathematics Subject Classification: 13C05, 13C13