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: ''This article addresses the notion of quasiregularity in the context of [[ring theory]], a branch of [[Abstract algebra|modern algebra]]. For other notions of quasiregularity in [[mathematics]], see the disambiguation page [[quasiregular (disambiguation)|quasiregular]].''
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In [[mathematics]], specifically [[ring theory]], the notion of '''quasiregularity''' provides a computationally convenient way to work with the [[Jacobson radical]] of a ring.<ref name="Isaacs, p. 180">Isaacs, p. 180</ref> Intuitively, '''quasiregularity''' captures what it means for an element of a ring to be "bad"; that is, have undesirable properties.<ref>Isaacs, p. 179</ref> Although a "bad element" is necessarily quasiregular, quasiregular elements need not be "bad," in a rather vague sense. In this article, we primarily concern ourselves with the notion of quasiregularity for [[unital ring]]s. However, one section is devoted to the theory of quasiregularity in non-unital rings, which constitutes an important aspect of noncommutative ring theory.
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==Definition==
 
Let ''R'' be a ring (with [[multiplicative identity|unity]]) and let ''r'' be an element of ''R''. Then ''r'' is said to be '''quasiregular''', if 1 - ''r'' is a [[Unit (ring theory)|unit]] in ''R''; that is, invertible under multiplication.<ref name="Isaacs, p. 180"/>  The notions of '''right or left quasiregularity''' correspond to the situations where 1 - ''r'' has a right or left inverse, respectively.<ref name="Isaacs, p. 180"/>
 
An element ''x'' of a non-unital ring is said to be '''right quasiregular''' if there is ''y'' such that <math>xy + x + y = 0</math>.<ref>Kaplansky, p. 85</ref> The notion of a '''left quasiregular''' element is defined in an analogous manner. The element ''y'' is sometimes referred to as a '''right quasi-inverse''' of ''x''.<ref>Polcino & Sehgal (2002), {{Google books quote|id=7m9P9hM4pCQC|page=298|text=this element is called a left quasi inverse of x|p. 298}}.</ref> If the ring is unital, this definition quasiregularity coincides with that given above. If one writes <math>x \cdot y = xy + x + y</math>, then this binary operation <math>\cdot</math> is associative. Therefore, if an element possesses both a left and right quasi-inverse, they are equal.<ref>Since ''0'' is the multiplicative identity, if <math>x \cdot y = 0 = y' \cdot x</math>, then <math>y = (y' \cdot x) \cdot y = y' \cdot (x \cdot y) = y'</math>. Quasiregularity does not require the ring to have a multiplicative identity.
</ref>
 
 
== Examples ==
 
* If ''R'' is a ring, then the additive identity of ''R'' is always quasiregular
* If <math>-x^2</math> is right (resp. left) quasiregular, then <math>x</math> is right (resp. left) quasiregular.<ref>Kaplansky, p. 108</ref>
* If ''R'' is a ring, every [[nilpotent element]] of ''R'' is quasiregular. This fact is also supported by an elementary computation:
 
:If <math>x^n = 0</math>, then
 
:<math>(1+x)(1 - x + x^2 - x^3 + ... + (-x)^n) = 1</math>
 
* A matrix is quasiregular in a [[matrix ring]] if it does not possess -1 as an [[eigenvalue]]. More generally, a [[bounded operator]] is quasiregular if -1 is not in its spectrum.
* In a unital Banach algebra, if <math>\|x\| < 1</math>, then the geometric series <math>\sum_0^\infty (-x)^n</math> converges. Consequently, every such ''x'' is quasiregular.
* If ''R'' is a ring and ''S''=''R''[[''X''<sub>1</sub>,...,''X''<sub>n</sub>]] denotes the ring of [[formal power series]] in ''n'' intederminants over ''R'', an element of ''S'' is quasiregular if and only its constant term is quasiregular as an element of ''R''.
 
==Properties==
 
* Every element of the [[Jacobson radical]] of a (not necessarily commutative) ring is quasiregular.<ref>Isaacs, Theorem 13.4(a), p. 180</ref> In fact, the Jacobson radical of a ring can be characterized as the unique right ideal of the ring, maximal with respect to the property that every element is right quasiregular.<ref>Isaacs, Theorem 13.4(b), p. 180</ref><ref>Isaacs, Corollary 13.7, p. 181</ref> However, a right quasiregular element need not necessarily be a member of the Jacobson radical.<ref>Isaacs, p. 181</ref> This justifies the remark in the beginning of the article - "bad elements" are quasiregular, although quasiregular elements are not necessarily "bad." Elements of the Jacobson radical of a ring, are often deemed to be "bad."
 
* If an element of a ring is nilpotent and [[Center of a ring|central]], then it is a member of the ring's Jacobson radical.<ref>Isaacs, Corollary 13.5, p. 181</ref> This is because the [[Principal ideal|principal right ideal]] generated by that element consists of quasiregular (in fact, nilpotent) elements only.
 
* If an element, ''r'', of a ring is [[Idempotent element|idempotent]], it cannot be a member of the ring's Jacobson radical.<ref>Isaacs, Corollary 13.6, p. 181</ref> This is because idempotent elements cannot be quasiregular. This property, as well as the one above, justify the remark given at the top of the article that the notion of quasiregularity is computationally convenient when working with the Jacobson radical.<ref name="Isaacs, p. 180"/>
 
==Notes==
{{reflist|2}}
 
==References==
 
* {{cite book
| author = I. Martin Isaacs
| year = 1993
| title = Algebra, a graduate course
| edition = 1st
| publisher = Brooks/Cole Publishing Company
| isbn = 0-534-19002-2
}}
* {{cite book
| author = Irving Kaplansky
| year = 1969
| title = Fields and Rings
| publisher = The University of Chicago Press
}}
* {{cite book
| author = Milies, César Polcino
| author2 = Sehgal, Sudarshan K.
| year = 2002
| title = An introduction to group rings
| publisher = Springer
| isbn = 978-1-4020-0238-0
}}
 
==See also==
 
* [[Jacobson radical]]
* [[Nilradical of a ring|Nilradical]]
* [[Unit (ring theory)]]
* [[Nilpotent element]]
* [[Center of a ring]]
* [[Idempotent element]]
 
[[Category:Ring theory|*]]

Latest revision as of 23:10, 17 July 2014

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