associate

Note that this happens iff a divides b and b divides a.

Proof If a=bu, for some unit u then by definition b divides a. Since b=au^-1 we also see that a divides b.

On the other hand if we have a=bu and b=av, for some u,v elements of R then substituting we see that a=avu, that is a(1-vu)=0. Since R is an integral domain we deduce that a and b are associates.

For example, 3 and -3 are associate integers.

As*so"ci*ate (#), v. t. [imp. & p. p. Associated (#); p. pr. & vb. n. Associating (#).] [L. associatus, p. p. of associare; ad + sociare to join or unite, socius companion. See Social.]

1.

To join with one, as a friend, companion, partner, or confederate; as, to associate others with s in business, or in an enterprise.

2.

To join or connect; to combine in acting; as, particles of gold associated with other substances.

3.

To connect or place together in thought.

He succeeded in associating his name inseparably with some names which will last an long as our language. Macaulay.

4.

To accompany; to keep company with.

Friends should associate friends in grief and woe. Shak.

© Webster 1913.

As*so"ci*ate, v. i.

1.

To unite in company; to keep company, implying intimacy; as, congenial minds are disposed to associate.

2.

To unite in action, or to be affected by the action of a different part of the body.

E. Darwin.

© Webster 1913.

As*so"ci*ate (#), a. [L. associatus, p. p.]

1.

Closely connected or joined with some other, as in interest, purpose, employment, or office; sharing responsibility or authority; as, an associate judge.

While I descend . . . to my associate powers. Milton.

2.

Admitted to some, but not to all, rights and privileges; as, an associate member.

3. Physiol.

Connected by habit or sympathy; as, associate motions, such as occur sympathetically, in consequence of preceding motions.

E. Darwin.

© Webster 1913.

As*so"ci*ate, n.

1.

A companion; one frequently in company with another, implying intimacy or equality; a mate; a fellow.

2.

A partner in interest, as in business; or a confederate in a league.

3.

One connected with an association or institution without the full rights or privileges of a regular member; as, an associate of the Royal Academy.

4.

Anything closely or usually connected with another; an concomitant.

The one [idea] no sooner comes into the understanding, than its associate appears with it. Locke.

Syn. -- Companion; mate; fellow; friend; ally; partner; coadjutor; comrade; accomplice.

© Webster 1913.