Two elements in a ring with unity are associates or associated elements of each other if one can be obtained from the other by multiplying by some unit, that is, a and b are associates if there is a unit u such that  a=bu.  Equivalently, one can say that two associates are divisible by each other.

The binary relationMathworldPlanetmath “is an associate of” is an equivalence relationMathworldPlanetmath on any ring with unity. For example, the equivalence classMathworldPlanetmath of the unity of the ring consists of all units of the ring.

Examples. In the ring of the rational integers, only opposite numbers ±n are associates.  Among the polynomials, the associates of a polynomial are gotten by multiplying the polynomial by an element belonging to the coefficient ring in question (and being no zero divisorMathworldPlanetmath).

In an integral domainMathworldPlanetmath, two elements are associates if and only if they generate the same principal idealMathworldPlanetmathPlanetmathPlanetmathPlanetmath.

Title associates
Canonical name Associates
Date of creation 2013-03-22 11:56:31
Last modified on 2013-03-22 11:56:31
Owner drini (3)
Last modified by drini (3)
Numerical id 10
Author drini (3)
Entry type Definition
Classification msc 13-00
Classification msc 16-00
Related topic Ring
Related topic Unit
Defines associate
Defines associated element