Stable subsets of modules and the existence of a unit in associative rings

We obtain criteria for the existence of a (left) unit in rings (arbitrary, Artinian, Noetherian, prime, and so on) that are based on the systematic study of properties of stable subsets of modules and their stabilizers that generalize the technique of idempotents. We study a class of quasiunitary rings that is a natural extension of classes of rings with unit and of von Neumann (weakly) regular rings, which inherits may properties of these classes. Some quasiunitary radicals of arbitrary rings are constructed, and the size of these radicals “measures the probability” of the existence of a unit. Translated fromMatematicheskie Zametki, Vol. 61, No. 4, pp. 596–611, April, 1997. Translated by A. I. Shtern     

New results on the peak algebra

The peak algebra is a unital subalgebra of the symmetric group algebra, linearly spanned by sums of permutations with a common set of peaks. By exploiting the combinatorics of sparse subsets of [n−1] (and of certain classes of compositions of n called almost-odd and thin), we construct three new linear bases of . We discuss two peak analogs of the first Eulerian idempotent and construct a basis of semi-idempotent elements for the peak algebra. We use these bases to describe the Jacobson radical of and to characterize the elements of in terms of the canonical action of the symmetric groups on the tensor algebra of a vector space. We define a chain of ideals of , j = 0,..., , such that is the linear span of sums of permutations with a common set of interior peaks and is the peak algebra. We extend the above results to , generalizing results of Schocker (the case j = 0). Aguiar supported in part by NSF grant DMS-0302423 Orellana supported in part by the Wilson Foundation     

Some reducibility properties for pseudovarieties of the form DRH

Let H be a pseudovariety of groups and DRH be the pseudovariety containing all finite semigroups whose regular ?-classes belong to H. We study the relationship between reducibility of H and of DRH with respect to several particular classes of systems of equations. The classes of systems considered (of pointlike, idempotent pointlike and graph equations) are known to play a role in decidability questions concerning pseudovarieties of the forms V?W, V∨W, and V W.     

Homological behavior of idempotent subalgebras and Ext algebras

Let A be a(left and right) Noetherian ring that is semiperfect. Let e be an idempotent of A and consider the ring Γ :=(1-e)A(1-e) and the semi-simple right A-module Se := e A/e rad A. In this paper, we investigate the relationship between the global dimensions of A and Γ, by using the homological properties of Se. More precisely, we consider the Yoneda ring Y(e) := Ext_A~*(Se, Se) of e. We prove that if Y(e) is Artinian of finite global dimension, then A has finite global dimension if and only if so does Γ. We also investigate the situation where both A and Γ have finite global dimension. When A is Koszul and finite dimensional, this implies that Y(e) has finite global dimension. We end the paper with a reduction technique to compute the Cartan determinant of Artin algebras. We prove that if Y(e) has finite global dimension, then the Cartan determinants of A and Γ coincide. This provides a new way to approach the long-standing Cartan determinant conjecture.     

On Fully Idempotent Modules

A submodule N of a module M is idempotent if N = Hom(M, N)N. The module M is fully idempotent if every submodule of M is idempotent. We prove that over a commutative ring, cyclic idempotent submodules of any module are direct summands. Counterexamples are given to show that this result is not true in general. It is shown that over commutative Noetherian rings, the fully idempotent modules are precisely the semisimple modules. We also show that the commutative rings over which every module is fully idempotent are exactly the semisimple rings. Idempotent submodules of free modules are characterized.     

Generalized quadraticity of linear combination of two generalized quadratic matrices

It is investigated the necessary and sufficient conditions for the generalized quadraticity of a linear combination of any two generalized quadratic matrices. The main result obtained is, in a sense, a generalization of the main results given in [Uç M, Özdemir H, Özban AY. On the quadraticity of linear combinations of quadratic matrices. Linear Multilinear Algebra. 2015;63:1125–1137.] which contains many of the results in the literature related to idempotency or involutivity of the linear combinations of idempotent and/or involutive matrices, to the generalized quadratic matrices.     

On the spectrum of linear combinations of two projections in C*-algebras

In this note, we study the spectrum and give estimations for the spectral radius of linear combinations of two projections in C*-algebras. We also study the commutator of two projections.     

The group inverse of the combinations of two idempotent matrices

In this article, we discuss the group inverse of aP + bQ + cPQ + dQP + ePQP + fQPQ + gPQPQ of idempotent matrices P and Q, where a, b, c, d, e, f, g ∈ ? and a ≠ 0, b ≠ 0, put forward its explicit expressions, and some necessary and sufficient conditions for the existence of the group inverse of aP + bQ + cPQ.