Baer ring

Baer Rings

Let R be a ring with multiplicative identityPlanetmathPlanetmath 1. Then R is called a left Baer ring if, for any subset S of R, the left annihilator of S is left principal, generated by an idempotentPlanetmathPlanetmath:


A right Baer ring is defined similarly, by replacing the word left with right above. It turns out that a left Baer ring is a right Baer ring, and vice versa, so we may drop the word left or right in the name.

Clearly, a domain is a Baer ring. And, by Wedderburn-Artin theorem, a semisimple ringPlanetmathPlanetmath is also a Baer ring. Another example, found in operator theory, is the ring of bounded linear operators on a Hilbert spaceMathworldPlanetmath.

A closely related concept is that of a left (right) Baer *-ring: it is a ring with involution * such that the left (right) annihilator of any subset is left (right) principal, generated by a projection (an idempotent that is in addition a self-adjoint element).

A left Baer *-ring is a left Baer ring, a right Baer *-ring is a right Baer ring. And, interestingly, the notion of left and right is also redundent for Baer *-rings. For example, let’s show left means right. Since a left Baer * ring R is left, and consequently right Baer, the right annihilator of a subset S has the form r.ann(S)=eR for some idempotent eR. Since Se=0, e*S*=0 or that l.ann(S*)=Re*. But R is left Baer *, l.ann(S*)=Rf for some projection fR. So fR=f*R*=(Rf)*=(Re*)*=e**R*=eR=r.ann(S).

Rickart Rings

A closely related type of rings is called a left Rickart ring. A ring is left Rickart if the left annihilator of any element aR is left principal, generated by an idempotent:


where e=ee. A right Rickart ring is similarly defined. Clearly, a Baer ring is both right and left Rickart. However, the converseMathworldPlanetmath is not always true. In fact, right Rickart and left Rickart are not the same. An example of a left Rickart ring that is not right Rickart can be found in the second reference below. It may be shown that if all idempotents are central (lying in the center) in a left Rickart ring R, then R is also right Rickart (and vice versa).


  • 1 I. Kaplansky, Rings of Operators, W. A. Benjamin, Inc., New York, 1968.
  • 2 T. Y. Lam, Lectures on Modules and Rings, Springer, New York, 1998.
Title Baer ring
Canonical name BaerRing
Date of creation 2013-03-22 15:52:07
Last modified on 2013-03-22 15:52:07
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Definition
Classification msc 47C10
Classification msc 47C15
Classification msc 16U99
Defines Baer *-ring
Defines Rickart ring