# Derivations on Pseudoquotients

### Abstract

A space of pseudoquotients denoted by*B*(

*X, S*) is defined as equivalence classes of pairs (

*x*,

*f*); where

*x*is an element of a nonempty set

*X*,

*f*is an element of

*S*; a commutative semigroup of injective maps from

*X*to

*X*; and (

*x*,

*f*) ~ (

*y*,

*g*) for

*gx*=

*fy*: If

*X*is a ring and elements of S are ring homomorphisms, then

*B*(

*X*,

*S*) is a ring. We show that, under natural conditions, a derivation on

*X*has a unique extension to a derivation on

*B*(

*X, S*): We also consider (

*α*,

*β*) -Jordan derivations, inner derivations, and generalized derivations.

Published

25.06.2013

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 65, no. 6, June 2013, pp. 863–869, http://umj.imath.kiev.ua/index.php/umj/article/view/2472.

Issue

Section

Short communications