Related papers: Peirce decompositions, idempotents and rings
In this paper, we investigate the structural and characterizing properties of the so-called {\it 2-UQ rings}, that are rings such that the square of every unit is the sum of an idempotent and a quasi-nilpotent element that commute with each…
The theory of generalized inverses of matrices and operators is closely connected with projections, i.e., idempotent (bounded) linear transformations. We show that a similar situation occurs in any associative ring $\mathcal{R}$ with a unit…
This book is a rigorous and conceptually oriented introduction to ring theory. The emphasis is on structural understanding rather than encyclopedic coverage: rings are studied through ideals, homomorphisms, quotients, and universal…
We give a uniform description of the decomposition of the unipotent variety of a classical group in arbitrary characteristic into pieces (considered in a non-uniform way in the earlier parts of this paper).
We define a ring whose elements are rational functions, whose addition is polynomial multiplication, and whose multiplication is a convolution operation. It is then show that this ring's endomorphisms exhibit a strong classification.…
The article focuses on four aspects related to the descent algebras of type $A$. They are modular idempotents, higher Lie powers, higher Lie modules and the right ideals of the symmetric group algebras generated by the Solomon's descent…
In this paper, we are mainly interested in the two questions "which are the commutative rings on which every finitely presented modules is [Formula: see text]-periodic (respectively, [Formula: see text]-periodic)?". It is proved that these…
Continuing the study of the structure of semirings, we turn to the spectrum of prime congruences. Joo and Mincheva developed an elegant theory in the special case of idempotent semirings, which is generalized here to ``semiring pairs,''…
We consider and study those rings in which each nil-clean or clean element is uniquely nil-clean. We establish that, for abelian rings, these rings have a satisfactory description and even it is shown that the classes of abelian rings and…
The set of all subsets of any inverse semigroup forms an involution semiring under set-theoretical union and element-wise multiplication and inversion. We find structural conditions on a finite inverse semigroup guaranteeing that neither…
The separability tensor element of a separable extension of noncommutative rings is an idempotent when viewed in the correct endomorphism ring; so one speaks of a separability idempotent, as one usually does for separable algebras. It is…
We explain a derived version of the basic construction of localisations of module categories by means of idempotent ideals, which lie at the heart of Faltings' almost ring theory. We use it to provide an example of a commutative algebra in…
This is a survey. The main subject of this survey is the homotopical or homological nature of certain structures which appear in classical problems about groups, Lie rings and group rings. It is well known that the (generalized) dimension…
The Dieudonn\'e crystal of a p-divisible group over a semiperfect ring R can be endowed with a window structure. If R satisfies a boundedness condition, this construction gives an equivalence of categories. As an application one obtains a…
Many classical ring-theoretic results state that an ideal that is maximal with respect to satisfying a special property must be prime. We present a "Prime Ideal Principle" that gives a uniform method of proving such facts, generalizing the…
An ordered semiring is a commutative semiring equipped with a compatible preorder. Ordered semirings generalise both distributive lattices and commutative rings, and provide a convenient framework to unify certain aspects of lattice theory…
We formulate and prove relative versions of several classical decompositions known in the theory of Chevalley groups over commutative rings. As an application we obtain upper estimates for the width of principal congruence subgroups in…
In this paper, we study the notion of a separability idempotent in the C*-algebra framework. This is analogous to the notion in the purely algebraic setting, typically considered in the case of (finite-dimensional) algebras with identity,…
We develop an extension of valuations theorem for suitable extensions of idempotent semirings. As an application, we give a new proof for the classical case of fields. Along the way, we develop characteristic one analogues of some central…
We classify the primitive idempotents of the $p$-local complex representation ring of a finite group $G$ in terms of the cyclic subgroups of order prime to $p$ and show that they all come from idempotents of the Burnside ring. Our results…