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单词 Quasiregularity
释义

quasi-regularity


An element x of a ring is called right quasi-regular [resp. left quasi-regular] if thereis an element y in the ring such that x+y+xy=0 [resp. x+y+yx=0].

For calculations with quasi-regularity, it is useful to introduce the operationMathworldPlanetmath * defined:

x*y=x+y+xy.

Thus x is right quasi-regular if there is an element y such that x*y=0.The operation * is easily demonstrated to be associative, and x*0=0*x=x for all x.

An element x is called quasi-regular if it is both left and right quasi-regular.In this case, there are elements y and z such that x+y+xy=0=x+z+zx(equivalently, x*y=z*x=0).A calculation shows that

y=0*y=(z*x)*y=z*(x*y)=z.

So y=z is a unique element, depending on x, called the quasi-inverse of x.

An ideal (one- or two-sided) of a ring is called quasi-regular if each of its elements is quasi-regular. Similarly, a ring is called quasi-regular if each of its elements is quasi-regular (such rings cannot have an identity elementMathworldPlanetmath).

Lemma.

Let A be an ideal (one- or two-sided) in a ring R.If each element of A is right quasi-regular, then A is a quasi-regular ideal.

This lemma means that there is no extra generality gained in defining terms such as right quasi-regular left idealMathworldPlanetmathPlanetmath, etc.

Quasi-regularity is important because it provides elementary characterizationsMathworldPlanetmath of the Jacobson radicalMathworldPlanetmathfor rings without an identity element:

  • The Jacobson radical of a ring is the sum of all quasi-regular left (or right) ideals.

  • The Jacobson radical of a ring is the largest quasi-regular ideal of the ring.

For rings with an identity element, note that x is [right, left] quasi-regular if and only if 1+x is [right, left]invertible in the ring.

Titlequasi-regularity
Canonical nameQuasiregularity
Date of creation2013-03-22 13:12:59
Last modified on2013-03-22 13:12:59
Ownermclase (549)
Last modified bymclase (549)
Numerical id8
Authormclase (549)
Entry typeDefinition
Classificationmsc 16N20
Synonymquasi regular
Synonymquasi regularity
Related topicJacobsonRadical
Related topicRegularIdeal
Related topicHomotopesAndIsotopesOfAlgebras
Definesquasi-regular
Definesright quasi-regular
Definesleft quasi-regular
Definesquasi-inverse
Definesquasi-regular ideal
Definesquasi-regular ring
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更新时间:2025/5/4 2:58:12