In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element a such that a2 = a. That is, the element is...

Idempotent analysis Idempotent matrix Idempotent relation – a generalization of idempotence to binary relations Idempotent (ring theory) Involution (mathematics)...

additively idempotent semirings are zerosumfree and, indeed, the only additively idempotent semiring that has all additive inverses is the trivial ring and so...

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This...

a nonzero ring is necessarily a zero divisor. An idempotent e {\displaystyle e} is an element such that e2 = e. One example of an idempotent element is...

Boolean ring R is a ring for which x2 = x for all x in R, that is, a ring that consists of only idempotent elements. An example is the ring of integers...

is indecomposable if and only if its endomorphism ring does not contain any non-trivial idempotent elements. If the module is an injective module, then...

single-idempotent case in the first paragraph of this section. An idempotent of a ring is called central if it commutes with all elements of the ring. Two...

Using the ring R as above, with residue field K, the identity element of G may be decomposed as a sum of mutually orthogonal primitive idempotents (not necessarily...

Karoubi envelope (or Cauchy completion or idempotent completion) of a category C is a classification of the idempotents of C, by means of an auxiliary category...

mathematics, a clean ring is a ring in which every element can be written as the sum of a unit and an idempotent. A ring is a local ring if and only if it...

In algebra, an SBI ring is a ring R (with identity) such that every idempotent of R modulo the Jacobson radical can be lifted to R. The abbreviation SBI...

In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology...

an idempotent of R. Paralleling this example, von Neumann regular rings are seen to be both right and left Bézout rings. If D is a division ring and...

Rickart rings have also been termed left PP-rings. ("Principal implies projective": See definitions below.) An idempotent element of a ring is an element...

conditions, then R has no infinite pairwise orthogonal sets of idempotents. If R is a ring for which L A {\displaystyle {\mathcal {LA}}\,} satisfies the...

connected. Non-examples are given by product rings such as Z × Z; here the element (1, 0) is a non-trivial idempotent. In algebraic geometry, connectedness is...

zero-divisors, idempotents, and left- and right-identities) in Gregory Dresden's lecture notes. The occurrence of non-commutativity in finite rings was described...

homomorphism f : R → S maps any idempotent element to an idempotent element. If f : R → S is a rng homomorphism from a ring to a rng, and the image of f...

conjecture implies the idempotent conjecture and is implied by the unit conjecture. As of 2021, the zero divisor and idempotent conjectures are open. The...

of BP. For each prime p, Daniel Quillen showed there is a unique idempotent map of ring spectra ε from MUQ(p) to itself, with the property that ε([CPn])...

\mathbb {C} \otimes _{\mathbb {R} }\mathbb {C} } . This ring has two non-trivial idempotents, e 1 = 1 2 ( 1 ⊗ 1 ) − 1 2 ( i ⊗ i ) {\displaystyle e_{1}={\tfrac...

thus called a group Hopf algebra. The apparatus of group rings is especially useful in the theory of group representations. Let G {\displaystyle G} be a...

is a morphism g : Y → X such that g ∘ f = idX. Thus f ∘ g : Y → Y is idempotent; that is, (f ∘ g)2 = f ∘ (g ∘ f) ∘ g = f ∘ g. The left inverse g is also...

correspondence between idempotents and maximal subgroups. Here the term maximal subgroup differs from its standard use in group theory. More can often be...

module into submodules is the same as to give orthogonal idempotents in the endomorphism ring of the module that sum up to the identity map. Indeed, if...

Annihilator (disambiguation) Annihilator (ring theory) – Ideal that maps to zero a subset of a module Idempotent (ring theory) – In mathematics, element that equals...

Noetherian commutative ring R, then R / I {\displaystyle R/I} is not a flat module, except if I is generated by an idempotent (that is an element equal...

localization of C in the sense of an idempotent and coaugmented functor. Serre introduced the idea of working in homotopy theory modulo some class C of abelian...

mathematics, a Zorn ring is an alternative ring in which for every non-nilpotent x there exists an element y such that xy is a non-zero idempotent (Kaplansky 1968...