Topic
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.
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TL;DR: The main objective in this paper is to prove that any multiplicative derivation of R is additive if and only if d is additive, i.e., if the derivation is additive.
Abstract: Our main objective in this note is to prove the following Suppose R is a
ring having an idempotent element e ( e ≠ 0 , e ≠ 1 ) which satisfies:
( M 1 ) x R = 0 implies x = 0 ( M 2 ) e R x = 0 implies x = 0 ( and hence R x = 0 implies x = 0 ) ( M 3 ) e x e R ( 1 − e ) = 0 implies e x e = 0
If d is any multiplicative derivation of R , then d is additive
98 citations
TL;DR: Theorem Theorem as mentioned in this paper states that a congruence class, C, of a semilattice is closed, convex, and contains a maximal element, i.e., the product p of all elements in C is in C, and by idempotence pc=p for an arbitrary ceC.
Abstract: Appendix II: A Theorem Theorem: A congruence class, C, of a semilattice is closed, convex, and contains a maximal element. (1) C is closed. Assume aEC and beC. Then abeCC. Now,a=aacCC, so C=CC. (2) C contains a maximal element. By closure the product, p, of all elements in C is in C, and by idempotence pc=p for an arbitrary ceC. Hence p is a maximal element of C. (3) C is convex. Let aeC, beC, a>x>b, xeX. xa-a so XC=C, xb-x so XC-X. Thus X =C.
97 citations
TL;DR: It is shown that sup- and inf-decomposable measures can be obtained as limits of families of Pseudo-additive measures with respect to generated pseudo-additions as well as corresponding integrals of g-integrals.
94 citations
28 Jun 2010
TL;DR: The characterization of idempotent uninorms is revisited and some technical aspects are corrected and the same characterization is translated in terms of symmetrical functions.
Abstract: In this paper the characterization of idempotent uninorms given in [21] is revisited and some technical aspects are corrected Examples clarifying the situation are given and the same characterization is translated in terms of symmetrical functions The particular cases of left-continuity and right-continuity are studied retrieving the results in [7]
92 citations
TL;DR: In this paper, the classical theory of Morita equivalence is extended to idempotent rings which do not necessarily have an identity element, and the role of progenerators is played by the unital and codivisible modules.
91 citations