Idempotence

About: Idempotence is a research topic. Over the lifetime, 1860 publications have been published within this topic receiving 19976 citations. The topic is also known as: idempotent.

Prime (-1,1) rings with idempotent

Abstract: Maneri [7] proved that a simple ring of type (-1, 1) with characteristic prime to 6 having an idempotent e#O, 1 is associative. It is shown in this paper that when R is a (-1, 1) ring with no trivial ideals which has characteristic prime to 6, then if R contains an idempotent e $0O, 1, it has a Peirce decomposition relative to e. Further, the multiplicative relations between the submodules of the Peirce decomposition relative to containment are the same as those for an associative ring. Under the additional assumption that R is a prime ring it is proven that R must be associative.

The tensor product of function semimodules

Abstract: Given two domains of functions with values in a field, the canonical map from the algebraic tensor product of the vector spaces of functions on the two domains to the vector space of functions on the product of the two domains is well known to be injective, but not generally surjective. By constructing explicit examples, we show that the corresponding map for semimodules of semiring-valued functions is in general not even injective. This impacts the formulation of topological quantum field theories over semirings. We also confirm the failure of surjectivity for functions with values in complete, additively idempotent semirings by describing a large family of functions that do not lie in the image.

The subvariety lattice for representable idempotent commutative residuated lattices

Abstract: RICRL denotes the variety of commutative residuated lattices which have an idempotent monoid operation and are representable in the sense that they are subdirect products of linearly ordered algebras. It is shown that the subvariety lattice of RICRL is countable, despite its complexity and in contrast to several varieties of closely related algebras.

On modules for which the finite exchange property implies the countable exchange property

Abstract: It has been a long standing open problem whether the finite exchange property implies the full exchange property for an arbitrary module The main results of this paper are Theorem 11: For modules whose idempotent endomorphisms are central, the finite exchange property implies the countable exchange property, and Theorem 211: Over a ring with ace on essential right ideals, the finite exchange property implies the full exchange property for every quasi-continuous module The latter can be viewed as a partial affirmative answer to an open problem of Mohamed and Muller [8]

Auslander's formula and correspondence for exact categories

Abstract: The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ of admissibly finitely presented functors and use it to give a version of Auslander correspondence for any exact category $\mathcal{E}$. An important ingredient in the proof is the localization theory of exact categories. We also investigate how properties of $\mathcal{E}$ are reflected in $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$, for example being (weakly) idempotent complete or having enough projectives or injectives. Furthermore, we describe $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ as a subcategory of $\operatorname{mod}(\mathcal{E})$ when $\mathcal{E}$ is a resolving subcategory of an abelian category. This includes the category of Gorenstein projective modules and the category of maximal Cohen-Macaulay modules as special cases. Finally, we use $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ to give a bijection between exact structures on an idempotent complete additive category $\mathcal{C}$ and certain resolving subcategories of $\operatorname{mod}(\mathcal{C})$.